LeetCode #1044 — HARD

Longest Duplicate Substring

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

Given a string s, consider all duplicated substrings: (contiguous) substrings of s that occur 2 or more times. The occurrences may overlap.

Return any duplicated substring that has the longest possible length. If s does not have a duplicated substring, the answer is "".

Example 1:

Input: s = "banana"
Output: "ana"

Example 2:

Input: s = "abcd"
Output: ""

Constraints:

2 <= s.length <= 3 * 10⁴
s consists of lowercase English letters.

Step 01

Brute Force Baseline

Problem summary: Given a string s, consider all duplicated substrings: (contiguous) substrings of s that occur 2 or more times. The occurrences may overlap. Return any duplicated substring that has the longest possible length. If s does not have a duplicated substring, the answer is "".

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Binary Search · Sliding Window

Example 1

"banana"

Example 2

"abcd"

Step 02

Core Insight

What unlocks the optimal approach

Binary search for the length of the answer. (If there's an answer of length 10, then there are answers of length 9, 8, 7, ...)
To check whether an answer of length K exists, we can use Rabin-Karp 's algorithm.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1044: Longest Duplicate Substring
class Solution {
    private long[] p;
    private long[] h;

    public String longestDupSubstring(String s) {
        int base = 131;
        int n = s.length();
        p = new long[n + 10];
        h = new long[n + 10];
        p[0] = 1;
        for (int i = 0; i < n; ++i) {
            p[i + 1] = p[i] * base;
            h[i + 1] = h[i] * base + s.charAt(i);
        }
        String ans = "";
        int left = 0, right = n;
        while (left < right) {
            int mid = (left + right + 1) >> 1;
            String t = check(s, mid);
            if (t.length() > 0) {
                left = mid;
                ans = t;
            } else {
                right = mid - 1;
            }
        }
        return ans;
    }

    private String check(String s, int len) {
        int n = s.length();
        Set<Long> vis = new HashSet<>();
        for (int i = 1; i + len - 1 <= n; ++i) {
            int j = i + len - 1;
            long t = h[j] - h[i - 1] * p[j - i + 1];
            if (vis.contains(t)) {
                return s.substring(i - 1, j);
            }
            vis.add(t);
        }
        return "";
    }
}

// Accepted solution for LeetCode #1044: Longest Duplicate Substring
func longestDupSubstring(s string) string {
	base, n := 131, len(s)
	p := make([]int64, n+10)
	h := make([]int64, n+10)
	p[0] = 1
	for i := 0; i < n; i++ {
		p[i+1] = p[i] * int64(base)
		h[i+1] = h[i]*int64(base) + int64(s[i])
	}
	check := func(l int) string {
		vis := make(map[int64]bool)
		for i := 1; i+l-1 <= n; i++ {
			j := i + l - 1
			t := h[j] - h[i-1]*p[j-i+1]
			if vis[t] {
				return s[i-1 : j]
			}
			vis[t] = true
		}
		return ""
	}
	left, right := 0, n
	ans := ""
	for left < right {
		mid := (left + right + 1) >> 1
		t := check(mid)
		if len(t) > 0 {
			left = mid
			ans = t
		} else {
			right = mid - 1
		}
	}
	return ans
}

# Accepted solution for LeetCode #1044: Longest Duplicate Substring
class Solution:
    def longestDupSubstring(self, s: str) -> str:
        def check(l):
            vis = set()
            for i in range(n - l + 1):
                t = s[i : i + l]
                if t in vis:
                    return t
                vis.add(t)
            return ''

        n = len(s)
        left, right = 0, n
        ans = ''
        while left < right:
            mid = (left + right + 1) >> 1
            t = check(mid)
            ans = t or ans
            if t:
                left = mid
            else:
                right = mid - 1
        return ans

// Accepted solution for LeetCode #1044: Longest Duplicate Substring
struct Solution;

use std::collections::HashMap;

const MOD: u64 = 1_000_000_007;
const P: u64 = 26;

impl Solution {
    fn longest_dup_substring(s: String) -> String {
        let mut lo = 0;
        let mut hi = s.len();
        let mut res = "".to_string();
        let s: Vec<char> = s.chars().collect();
        let n = s.len();
        while lo < hi {
            let mid = (lo + hi + 1) / 2;
            if let Some(s) = Self::exist(mid, &s, n) {
                res = s;
                lo = mid;
            } else {
                hi = mid - 1;
            }
        }
        res
    }

    fn exist(size: usize, s: &[char], n: usize) -> Option<String> {
        let mut pos: HashMap<u64, usize> = HashMap::new();
        let mut hash: u64 = 0;
        let mut pn: u64 = 1;
        for i in 0..size {
            hash *= P;
            hash += (s[i] as u8 - b'a') as u64;
            hash %= MOD;
            pn *= P;
            pn %= MOD;
        }
        pos.insert(hash, size);
        for i in size..n {
            hash *= P;
            hash += (s[i] as u8 - b'a') as u64;
            hash += MOD;
            hash -= (pn * (s[i - size] as u8 - b'a') as u64) % MOD;
            hash %= MOD;
            if let Some(end) = pos.insert(hash, i + 1) {
                if s[end - size..end] == s[i + 1 - size..=i] {
                    return Some(s[i + 1 - size..=i].iter().copied().collect::<String>());
                }
            }
        }
        None
    }
}

#[test]
fn test() {
    let s = "banana".to_string();
    let res = "ana".to_string();
    assert_eq!(Solution::longest_dup_substring(s), res);
    let s = "abcd".to_string();
    let res = "".to_string();
    assert_eq!(Solution::longest_dup_substring(s), res);
    let s = "ababdaebdabedeabbdddbcebaccececbccccebbcaaabaadcadccddaedaacaeddddeceedeaabbbbcbacdaeeebaabdabdbaebadcbdebaaeddcadebedeabbbcbeadbaacdebceebceeccddbeacdcecbcdbceedaeebdaeeabccccbcccbceabedaedaacdbbdbadcdbdddddcdebbcdbcabbebbeabbdccccbaaccbbcecacaebebecdcdcecdeaccccccdbbdebaaaaaaeaaeecdecedcbabedbabdedbaebeedcecebabedbceecacbdecabcebdcbecedccaeaaadbababdccedebeccecaddeabaebbeeccabeddedbeaadbcdceddceccecddbdbeeddabeddadaaaadbeedbeeeaaaeaadaebdacbdcaaabbacacccddbeaacebeeaabaadcabdbaadeaccaecbeaaabccddabbeacdecadebaecccbabeaceccaaeddedcaecddaacebcaecebebebadaceadcaccdeebbcdebcedaeaedacbeecceeebbdbdbaadeeecabdebbaaebdddeeddabcbaaebeabbbcaaeecddecbbbebecdbbbaecceeaabeeedcdecdcaeacabdcbcedcbbcaeeebaabdbaadcebbccbedbabeaddaecdbdbdccceeccaccbdcdadcccceebdabccaebcddaeeecddddacdbdbeebdabecdaeaadbadbebecbcacbbceeabbceecaabdcabebbcdecedbacbcccddcecccecbacddbeddbbbadcbdadbecceebddeacbeeabcdbbaabeabdbbbcaeeadddaeccbcdabceeabaacbeacdcbdceebeaebcceeebdcdcbeaaeeeadabbecdbadbebbecdceaeeecaaaedeceaddedbedbedbddbcbabeadddeccdaadaaeaeeadebbeabcabbdebabeedeeeccadcddaebbedadcdaebabbecedebadbdeacecdcaebcbdababcecaecbcbcdadacaebdedecaadbaaeeebcbeeedaaebbabbeeadadbacdedcbabdaabddccedbeacbecbcccdeaeeabcaeccdaaaddcdecadddabcaedccbdbbccecacbcdecbdcdcbabbeaacddaeeaabccebaaaebacebdcdcbbbdabadeebbdccabcacaacccccbadbdebecdaccabbecdabdbdaebeeadaeecbadedaebcaedeedcaacabaccbbdaccedaedddacbbbdbcaeedbcbecccdbdeddcdadacccdbcdccebdebeaeacecaaadccbbaaddbeebcbadceaebeccecabdadccddbbcbaebeaeadacaebcbbbdbcdaeadbcbdcedebabbaababaacedcbcbceaaabadbdcddadecdcebeeabaadceecaeccdeeabdbabebdcedceaeddaecedcdbccbbedbeecabaecdbba".to_string();
    let res = "aeeebaabd".to_string();
    assert_eq!(Solution::longest_dup_substring(s), res);
    let s = "shabhlesyffuflsdxvvvoiqfjacpacgoucvrlshauspzdrmdfsvzinwdfmwrapbzuyrlvulpalqltqshaskpsoiispneszlcgzvygeltuctslqtzeyrkfeyohutbpufxigoeagvrfgkpefythszpzpxjwgklrdbypyarepdeskotoolwnmeibkqpiuvktejvbejgjptzfjpfbjgkorvgzipnjazzvpsjxjscekiqlcqeawsdydpsuqewszlpkgkrtlwxgozdqvyynlcxgnskjjmdhabqxbnnbflsscammppnlwyzycidzbhllvfvheujhnxrfujwmhwiamqplygaujruuptfdjmdqdndyzrmowhctnvxryxtvzzecmeqdfppsjczqtyxlvqwafjozrtnbvshvxshpetqijlzwgevdpwdkycmpsehxtwzxcpzwyxmpawwrddvcbgbgyrldmbeignsotjhgajqhgrttwjesrzxhvtetifyxwiyydzxdqvokkvfbrfihslgmvqrvvqfptdqhqnzujeiilfyxuehhvwamdkkvfllvdjsldijzkjvloojspdbnslxunkujnfbacgcuaiohdytbnqlqmhavajcldohdiirxfahbrgmqerkcrbqidstemvngasvxzdjjqkwixdlkkrewaszqnyiulnwaxfdbyianmcaaoxiyrshxumtggkcrydngowfjijxqczvnvpkiijksvridusfeasawkndjpsxwxaoiydusqwkaqrjgkkzhkpvlbuqbzvpewzodmxkzetnlttdypdxrqgcpmqcsgohyrsrlqctgxzlummuobadnpbxjndtofuihfjedkzakhvixkejjxffbktghzudqmarvmhmthjhqbxwnoexqrovxolfkxdizsdslenejkypyzteigpzjpzkdqfkqtsbbpnlmcjcveartpmmzwtpumbwhcgihjkdjdwlfhfopibwjjsikyqawyvnbfbfaikycrawcbkdhnbwnhyxnddxxctwlywjcisgqfsctzatdgqqauuvgclicdrpjcphysqdjaflpdbmvnhqggixxzcmpsysbwfkzwxzjictnngufpqhcxlbkodyrqlfomlkiefbmcfenugzqnyqqvgpxonmizkpjdlaqyyowjagzkzrzvcrupfyofeftyfvoqorzvxphhdhydnqiyiczfcgzsecxzsoaobwrixcajabjnvtoerzwayjowahrmuixmmkbtchogfizmvbjnpespxngxjxntohzatlpkcmpphmewevpteharnszafbpbexrvnbedieojezdhnyooiivhnhakilvkobxepbksnqrtxxuqhalvtjspyvporalbliiwjciamlhttaydhxoelimuorjnfvebjhcocbkrgbguwdncodskzzoqrzgavsbjcippetltqaxjhkqacwlgmsbxezqubyzeznnsoqegkykzlxohvitbmjcxllbrvgdijyovpjyeaojlyxqwnheyblznwoyikhqiutotpfukyqkvatxotulvlqzfcvskdccuixthzqrwymzccosjmjqjigehcnfphjuuybaxxukconatzseljyirycbhucxmwwftulfwfmyqyprlnsmxzyfmgjctgeunuuexhbrbsaaingqxqrjvpuhbvcmyztmkgenhonajrkzfrqjinjrbmjyinhwvlcmmxvbgvjgfmaoliosmxbonvlzoiqvkxxtoposygcgkcotohcrauivxxvmrghuauadwojxjligrgstczirnvhqpzwgjbvqzlkxltqnqrfxieggnuriytavbnouwhuamdtlspednyckixkhxedjmotiuucewllthhducwgwmgzxsbkqzfnqfynwisvsctyqdoaiypjivtxkxgoyhwhccklbdjoqykaqzljejlizgbehekmkfetvgfstmypmfnyoundudqlorcogbzoznddfalthwpmiewkmvogmzirbprjftbtffjrkrfminnechitfyfaujgtugadqbrskulsjbaunonxolauvsifevpdyurvfocxtkizflcuvltzuhwyhlbxaphifhtgkfktfnnmocpenrlujsuppbbuorvtubuiyszawzftijwhwgdyubjmmodzybiyunksuixnkariubegpdgctbayaynfskkuyhjvegsjwsbppodvhpjdjlzhxixswdncapxyfjspxeqxdfkhockvrzoisikaymoiqzqbjyoscwegfomlnurwboesfiszetebjblaolnovgvfcpnbemwambkhwcgdbhvkoluhjfxlfrfaeedocdilaboesauplmttewlbojkocklhsbzrtzeyhqtmgroupbzlymupmupsvlkzchclujuozzmngjvktzstsvocxrziuxelruwojzaleyrkjkdleavwqxwgjdbtiywqtdtaamrlcjogxufhgvoqpqkgopbtyqchzhexqgecheahjzxapqjdylzjqhzlzssbjmokncxalgasexztnlzfisxxpeerywlrjdohprewwnlwdbtwmfnnxnoolfhqqxzcvoymdbvmaoliedpvwzyvgyrwjguvoqnxrnaeqwvcfrqkwjmlvxovptultyfuxerelpfgctnpdxluqeruxkxqntosggfjqmrnlnkhhilznpycdrnemnktcsmzufpqgiraphzmgfhevzejhavsypohpttnnowfahpxfwmvxgwfuomxemdkzdlzldesmowzmhwoydnsovwykxqyllbmcurlvtwcfwxvvkxfknwwcwfjkzjtonalgijdsulcfagehiptszrcltbbypopdbmdfkyofelmrdmdbceguyxnkheqqtbletpqmjugpckmjyuuvsbqhyzmposwcgscnishluuhnwkyrkujefpgtsqrmcoortgitpdoagdncxlofkqozgngbtmlyyoyodcmcwetdtltupjrtemrjswekkfjvfecmvagyptjjuwsqpjwlxxosqhpssdvjraaicjfwvesyqfbumjjbqytkinpldxopxjzmvpigmberobyzyxwvwmlmbziduqhmbescgkvhgqtalmaxfsjlysmvrizgvrudstiwmaahtqehfbofvqwgqygvseykmgmhgjbxcrtdjqvojvyhohallyewqelzhjeuqmmsqhkluvqsfmxzbqqokehfoqrlqmwpnwojfowqpqebnuggeuvsszgfywceolvabyvbrwatuyherijsdqvpyyhdyradbammmchqkvdbxpbrxzrpfrsiiezvowrfqejibvociujtcwbygvfwojgfnvvwqlqqgipxhrogppzghtnweodaxuqxknnqnajlnsvheiycsvifvoljsncgnunsqcymnyoeeslrjflpprvtksimffvnuvakskdakvmlkpowfpfzdrcfctikhvvbagrvjlzjydnlmspzyynyjjfxnozpjjgjelipswsmfroitqphzsuqgumlnkxksbzhrsvcnfwufofhurmhksvvfjzggbtgrezkrkqmhduyqgwuwxoxaiifemtwrbilftiuhcgpjvqxldrnlzphdffncevlcyrxlpbwuswjfdegexeoooshdfqtqithpfocyowaqeedikssptyvkabhtaeotcwxccgguuotqvypugpcbwzalxwqbjdcokoxjnqhggpbbfeyjiellsikiqqtxpvzmjsfleepjpbxpeicxfcwbpprzgcrjgjaxshewradetsqsvfmcxptmksecfpynqzpctqpogcgokzrkltsbmwxkmynasohpkzjupapngusnvdjfqezqhyikllgkelewwwhhbdjvxdagnnxscjkotbbmhzkqbjwuwidrnvmztikmqjcxmcpgkoudhydmdvberfuvjnhlnfcsbpzmuquvrgogtfwefhqzkmxxgadtvjpxvurxprbsssihviypclwkjfaatzjxtvlzwaacqlwnqetgkldqaqghuihrgxbbpmjfsvaigqrhiiskkfibaeilqptkdsqqfwxeixuxgkiboaqnuaeutjcydnxyxnmattjrrxmthwvyipgazaxgrrjcvdnyxpktsldhluhicyqprxhljyfhawuvoonrwyklcdlmdvsgqrwqqomisksftsfyeifmupvylkjbagzyctuifbsrugqsbrkvskmundmczltpamhmgqespzgrkxebsvubrlmkwyqhjyljnkeqvdxtjxjvzlrubsiiahciwefwsswgssxmvyvgjrobvubcbgjomqajmotbcgqjudneovfbjtjzwqtsovzshmxeqofssukkvcdwlsdtyplrlgwtehnwvhhegtwkwnqqdiajpcaajsylesadaiflruewhrbrogbujbppunsqgytgnyuhnkejhccavaptbydtqhvyatftxcaaljyhhkkadzdhhzawgndunwwgknnbtqaddpszqgummmnomfqmdxqtwjexsbadfdqhnyixjslsaisscocbabivzokkgiinqqzsrtfpzjmxfkqmuzzlelqjtjidjarkwbwlcqrefokrlwdmuzyffdtajnqoimlzzpcgpjjwlqkusefzbgznhexzojxnzxmmedobgvdabtdoiskozrdrjscxwivaekrkyyfynuktmgyziteavdxfctvkfkrmsdwpaywzbkeojeycwdkectydojttisizruilwokhepscqdnjygiakomkhyujaffxjyxqmvkemqihpcdygprdeaxgjbkonfvgtzayfbmgwsskoyxjlknwwtehhhpjllhkcblyaxnbekoidbbyqvdqqsyfcemylmqskpxifcnhmemkkitqtbfwhmyemkzightkjbhlquticivpeeclpamsqoztxvdtcqbsonxyecnhcadtghkjckhrcdfggnqlwurydzbeybqkcfnnbwkciwaqdzgmcrbltvcuftxsqfpxnoombsbeoqxivgtkrjklxatgcorkdrvmngwlekeopgecefzxtprcoajoopxviijxilxfiwuajsbtcctfcqqgzhyjmonwdbyjlnneidyaqhhothzpzaxcthvbxpdcqofaeamxbqjwhunnqwclhcqhagawjsxygorgbuqryzickyplbaivkabbrkibqzqacabbwmnpndaqvbknbqcjuywxjrdbznndomwbbqfgulwczydrhrocebhygriiwxmwtjjyqqiqrjblxuamddlsiocdoysdaacuovljtpgocephnxuugdyggsnhcqiqhulyhlxiwzvhrtbmvjnhacwunckqzhpcthqgsupptdwkfeprbg".to_string();
    let res = "maoli".to_string();
    assert_eq!(Solution::longest_dup_substring(s), res);
    let s = "baaddcedaabeeebbddcdcdbaedcadbcbedddbbcbcbcdddbdddbdebecedabbdccdabbaaeedcbcacbcdebebaeeecaaddeceabbcaddabeceaceebedcabbcdcdbabdeeeacaceabaaaaeaeabbdecbdebedcedacbecccaeacbadecedbacbcdcebddebabbdddcdbbdccbdaccacaccbaabebcdbeadbdeebddbdddbabbeaeeabbdbcbcaaccdcabaebeadcaddeedcabebcbdbbbcdeeabbbadcbbadddadaddeacdbbbbecdadababacabcceaccdaddbcbbbccbdddacadcddddedeaeabebcdcaccceeecbdcaeebeabedbececeadbdcdcedadbbebedaaeacbedecddcabeddbadaaeceedbbedaadcbdebcbbbdcbeccaadeacdbebaedadadddbebccacecbcecdeeadaebbdbaeeaeebcadcbbedcdddacdeeaaeeeeebdccabbdaecdccaeacbecbdbadccaeeeeaceeabcacebebbccceadebeaeeeeceebaaaecbaccaccbccebeceaecdbbaacaadcecedddbcbacdaecbbddbaedbbbbbeacdeddaecbbdabebaaadceaabdedeadbbdbebbbcedbddaedccedcbebbdbbeaecdaabcdbadbbdbcdcbdeedbcbabebaaaeceebeccddcaaeebbedcdceeeebeaddcdabbcdbebdaabaedbeacccebbccbecaeeacbdaeeccdbdeadcdebbcddaeaebdacdadbecbacbaccdcaddbdcabadcdacadadbdbedadbecacecedccabbceeeeecaeccceedacbaeadacdcddccaeabbcecddddbbabaeeecedbaccabceecbbeedacbbcdddadccedbbcbcadccdddedeceaccdaeedcaebcdadbaededddbebaabdeebacabaaceabdeedabdacbeddadeaeedbddaccaacbababeeaeaecdbdaeabeeedbaeeddbcceecceebabbebebdedbaebbeebddeeccdabaeaecccbeeebacacdbecadabdcdeedeebccababdeeaccbdbbabdecdbdebabdbeeadacbccceebdeddebedbbbcbabdccbddeaceeccbecacdbbaecbbaabebadbbdbababddeccbcebcdbdbcebddaeaedeebacaaaecbdaabdedabcbedadcebbeadbcaadbcdbaeaacddeebcbebbdbecbcdebbcaaaacddebedabdaeeaeeaabbedcdabdceeaaddcbbeadccbcddceedcabbaecabadbadeabdbdcceaedcdbcdaeaebdbdbaacdeccedbaeeabbadaaaecceabceececaacecebabebbdebadeabdaadbbaebeeabebbdbeeccceccdbaceabbbeeecdccebdadeabbeddcdacdbeedbededbbedddabddeacbdbeeedcdeecaebcbabeecadccbedbeaadaebdcabedecdeacaccbeaadabddeadddccccbdcdecebeeceaadbaaabdedadbdccbdeabbdddebebcabbebaaabbbaccecebaecccaadecddbaabeebcabedebdabcbccaeadebabaacddecaccdacaadbbcdbdbdbeddedddebacaeadbaebaddedcebccdbbdcdbcaaaaccdaaceaebdadaceeeabedcacdddeeabecbacdcdbcebabccabecbbeeedbeeccabaeccccceaeaaceebcaacabbbcacdcebabdbcbaeeddaccdeadcbdedeabecdaecadbcbbdbcececbdbcdcbbbbdaacebccbaacbcaaeadbcbeeedaceebaeebadbbcceddbdaedabbdacdeadabeadccadbebcecbcdbaabbecddbaacaccbbbeaeadcebdccbaedaabdadcaebbdcdeaebeecdbbdbaadaedbeecccddacdedcdebbddaeecabebacaacdacdadcecdcaeaeaecbccedabbceebacdedceedeebbbacddacddedbaddaccaebeebeaadbeeeecdcaedadabddebcbcddccadaaadddbdbdccbdebbcededcdabbdbbdbbbedecadbdebebdedeebaadcaebebdececeaabbdaedeebbacecdbcbddeaaabdcbbdbabddabacbdebcbcccebdcccbcddcebdcbededbaadebebedcbebcccdddbabbeecbadcbeeeabbadddbbbcedcebbaabdeceecadaabdeaeabbebabbcedadbddcbedadcaededaedbeaaaeddadddabbeeaddddaaadedccddbcebeadcaebaaeabeeebecccbaddceebceeddabbdddcbeeadbecddebcebebeedcedbeabdbebcccdbbaadbabcbebadbecabbaeccbbeecaeebccdabbdcbadbdacbdccdccebeabceaaecbbeedeaacaddcedccaabaabeaecdadaceacacdcdcadabeacbeeabdcdaeceacdbdccccbbbaebcceaabcbecbabebcaebbbbdeaebbebaeeacbdbaddeaaeacedbdbaedecdbaecbbeddededbadbbadebabecdaadddbadeaabbeaebbaaeeacebcceabaaeabaecbdaeaccbedccceacdaceacceddaccdeddbaccbabdabdbcbeceaadcdaddbadeddbeceacbbebeacecccadcecacbecbacadbaaeaebbabceeddebbcbabbecbdbadcddddcebdabcadbbaeadbeadcceeccbacccbeccbdbaecbbdbeebbebbcbeccebccdcbbabcdaedbaaaaceeebddebdcebaecbadedaaccccabebdadbbcaceabbddeeeebcaebecaaaacaccaacddddcaedcdccbdabbbabbacdabedbbbaecccacadbacbcadabccedcababaedaceebcaccddeecdbcbbcdbbadbbbcccdcbdadbccaacbaeabbaecabccebcddbdbbbccbbcccdebbcabadcaeeecbacadaceeaecddeebeacadeabdabbbcdaeacdebddadbeeedbdbcebedbabbbebeaadabdabebaebaabdcbbceaececabecbdadeecccdebcbecaabbeccaedabbaaddeedaedbdbacdcaccbdbabbdebdebceceddbcddbabeebeecadcabbaedeabebdeeaaebaebaeaaaccbebceedadbcbbbaeceebcaedecbbebddccaadcadccbcababcadacbcabebcbbeecabbeecebeabceddaaeebdbdeeadeeebadbbcaaebcdcdeabaaecababaccaadcdebdeaddcaabaaaaedbadeddecdeaaecaedceabebdcceecdbcbcdaceeacececcadacaebadedebebcbcedcdbcadaddadbbadcebdeacceabaeedecbebbeccbabaaddddacccbddebbcbdedbebcccbbaecceeeccddacccaddecdcbaebdebbbccbdddeadaddacdcebebeedccddedddcbceadbbdddcabddbbbbdbabccbedcacabecabdeccdecebdbbcbdcdcaebbaceededeadcbccdbddbbebeeddbedaeadbdabebdcdbeecdaecbcecedcbeaceacdeebaadeeebaaeecdebeabadbbdbbcbeceebeeecdbdacebdcedbecccabdbaaacbeaecbddcaddbdeeeadeadedabacdeaadaedbdacccbbacadadedecddbcabcebdadeaecbeeaabacaddaeebcbdcabeacbddbecbbbbacbdbbdecebeedbeacdcabcdadcadcbbadaedaddabedbbbbabeddacedbaddcaececbdebeaeeceaecdedccbaadbdabdedbeccaeaddbaaabddbcabcbbbddbedbceddcdbcdbeeebbabbaedcdbdedaccacbddcdbaaadacdaabaeebdcedbdbedeeccaacbdcaaacbacccebcbadeececacbcaaaecccbabecedaebdaebdaecacaebceeddadecedbccdaababccadeeddbbdccecbddbbdacbeaabdaabebdaaebebcbacdccacebccaadaacaacbbaaaaadbdcebabddccebccebdbcbbebeadcbdcbaddcdaceedebdcdabaaeaeccdabadbdeacbaebcbcbadabadacdcebcdeaebddcabccddbababbadddaeaecbedcdeceebdaeebdbbdebabdaeeaedbcbdacdbbcbcaddeaaadcdbcdcacaabecbceabecbccaadaddaaeaccbbacceceddaedcebeccaaabcadaaddeacaebcdbbeeacbaaecddeadddcabbeebecdadbadceadecedccebbbcbdcbbceeedcbdcebaddabbdedceedeeaaabcbcecccadaebdcaeccabaaeebbbbaecbcbccbbacdadebcabbceeddcaebdccaeedcceeddecaaeebeeeadcacbacdaacbbbccdbdecbcaaceddaadacdcecbabdebdececeeecbedacddbaeaecbbcadeecbccccbcedcaddacdeecbaeaadeddcdcbabbabaeadecdcaaaaddbcddbabeeccbdbeaddbeceedbdbdcdcaabebcbacdbedbadbeecbbbdaabcceddcdabbdbdadbdcddeccbebdeaebbbaacceeddcbdadcacdaacebaeccdeacaddcbdbaeadcaaaceadcddeaaedadcadedaedacaaacabcdaaacecdbbabebebdeaadaecdcbbadeeecdbccedeebbddebecadcaeaeacacccecbadedaaeebddaeebbadcedaaadcdbcbcebabeccdddcbdaaadebeddbbdaeaceabecadcdecabdddbcedcdecceacbccbdbcaebbdbaabecedccaaceecbaabbebdeeaceedeadbaacdcebbbabdccaddcecedeebdeadebbacbcadbbcabccddeeccdcbdbcdacecaceddcdbaeececbdcaececdedabcaaaecdccebadedbdbbbdccececaeaebdacaddeacacddccdaccccadbecceaedacaacdabecdbbabecacabdebddcaadeeacccdbabebbbedadebaedeaaeeeaceaabdbccaabeabaaddbeaaadeabaddaabdcebdcebbedadacbeecddbcadbdccbabeeaecdebabdaabbcecdaebecebcaccadceeaaaaeeedbbaddeadacbccadbbeabeedacaddbaadabaebeaeeedbcbeeabebeeeedeaeadeccbbacadadbbceaacaecdcbbbaeacedbabcabddeacadcddaaaddabcccecbbcdaaeaccadbeaabbcdebcebdebdebabdbcdbdceadeccdabbeeaceebaadbdaadbcdeaeadbadcebeeabcbecaeaacbdceccbecadcceeabcabccadcddbdbcaabeadcdbbaeadacbbdddaedbccaadcceaacbaaaacbeaeaebdebdbdeaebcecaceeaebbebcadaadccbccdaedbbdbcceecacddeeadedbedaeddbaccddaecbeaadadcccabbeeabbecebdaacabedceadaebbbbccdccbabbeeebaaedcdecbaeaaaabdabebeacdcecaedbddbbccbcccceaedbeccdcdebaeaccbbdecaeeebcaebbdaacaedeebabecccdcaadecaeadeccedededeaeeddebedcdbeecedaaecdbacdddddeeebebacedabebbbeacbcddaeaeabbbcbcabdabcbdaedabbdbacedbebdcaeebecbacbeaceababecdcdaabaaeadebddeccdaabdececdaeceecabbbeeebceabbbbdebadbcccbcbdbbceeddbbddcbbceeeabbeedbddbdacedddcdabeecbddceabbccbcbededaaaacbeeaecaeccceddebeebeeacdeabbaaedcbedcebcbadaedbaddcaebadadccedeeabdeaeeaeaebaccabbeaebaedccaeadadbeaebbeceddeaecccaaadcabceadbedecddceaaedbeebeacaaebeeadceccddeebcecedaebaedacaedccbaebbedeeacecaceaaddaceaabdaeeeddacecabadabbdebccdccedcdaceeabcdeaeaedbbdeedbdcabecdcaabcaebeeabbbdeedaaecedaabbbaaecadbbbcebaeceaadccdbcaeaddccabcedaabecdeacebabbcdbbbacdbaaeebddeaaedcbccecbdaacaedccbaeaddaddaccccaceecadcadecbcecccabeccabaaceeccbeccbbdddcadbeeaeddcbbbddccacbbccbcdabbbdbbeddaaecddaecaacdcaaadaeccdbdbecededcdcaeadbdbedeeddeccbdbceddabececdaecbacabcedccceebbadeeabddabeceabaeecdaabcedacedbabcbeddddddebaaddecdacbebeadbbebbabeebcecacaebedebedeaeaddbbdbdcbcbcbadcccbddbebcacdebcedbbbacccdbaceacedabdacecdeccaaceaddbbdbeaeadcbadbeccbdccceddbcdbeccabcaaebbdaeccecdbbacbbdccbbededbdeeaeabeedbecadddcbdbcdceedecbaedbbcbabcadbbddabadcdbaeebcaaeeacbbbebbcaecedaebaecdeedcecebeebdabcbecdaecccecbbbbcbacebbaeaadabaadaebabdedcddcbcdddecdaaaaddcdedeceebbaddceeeadbbddaecaadabcedeabbcdddacbccdceeacddebaaacaccdcecbbcabeecdcbaabaececdbdebbcddeeadccddbecebbeaebbcbdddedbecccecacacabbaeababceddcdcaebebaadedddccecbebbbabbbeecddbbabbeeeeceadacadcaadcaedabdcdcacedaddececabbaeaadeebcccbeeeecbcaddacaceceedddabcccddbacbeeecbceeaeddcedaceebacddbcadbbeedaaecbbbecaeadbeaaaddeaeabdceddbbeaceaddaabcbabaecdcbcecdecdaeeeaccacddaaccdbbdeeeceedccbdeeecceaedecbeeaececaecaacbbbdcbcededbbbedbacdddcedbcaaebeaaebaeddeeeddabaddbcbcaeceedebcaacabeecedbacacedddbddcedcddbadadbdaccbbeebdbbaadadbcdeeaaebbacdbdedddcacaebbebbeeecaeacecbaaeccecbcebcdaadeeadcbddcebaeebabeddcddccacbacbceacaccbccbbcdaaebdcbacaebdaabebcaabaaebeddeaeacbcedbccbdebdccadaddacbebceaaabaebbddbcdadbcdcbdeaeecdadcbeabdadecedcdcdcdbeacddeacaaccebacaeddbbcbbdcaedadddebeddbcaeebbcdecbdaaaabcebdedacceadcccbaadedcceeccceabdbcbecccdbcbebcabbbcaebacdbedaaabcddbccadbeddccbeeceaeceacdecadedcabaeecedcbbbcaecdebbacdbdadaedaeacadbebecaaaaebadeedbdcebcecebadebeebddeecbacbadabdcceabebdbaaabaacacbebdddebcaebbbeecbdccbdcabdbddbbbadaceeaeecddcaabadeedeeebbdbbeaecbedbadcbadcecacacdaecbabadcceaadbdcbacbccdeabaabcdbdaaacdaedccbebdcbadccaddbccededeadaddaccaebabcbbdcaedaedccdacbaacecadcdbbaedbcddebadbdaedceebebacdceebeebaadaebeaeabdcbadbbaddbdabaececbaebbcdccedccbadaebdbeecaeedeeceeacdaaccecabeeaedacccccbaacaddacbbcbabbadcccaaabadcabbbadaeeaadacccacedaeecebccabcddeeadacdbaccebdabbaabadbecadadbbadeeebcbddaadececbcebbddebcebbbebccbdbedbdcebacedaeddeddaebcdbdbcbebabdbcbcbbeebaaeeeebcaceacdcdccaedabacddccbbbdabccadbcaaeeeabcbdeccbcbbddbbdaedcddabaccdacebacdccbbecaaeccadabddeaccbeeeecebdeecbceeddadadcacdecacacbddbdacbcdbadbdcddeedcceceecbaeecbcadacbbbabebceaadacceedbdcbcaedbbbaabadbceacccbbbdcbcebecbbabbdababcdbebeedebeabaabeaeaccdcdcecbdebeedbdbaaadbacdadcceeecdbababccbcbdeababbbbbeedbaaaaecccdacdeebdcaadbdcdbcccaabdbbaceceecdddecaeaddccceabadabbedbacdeddeebdceecedbabdcadeaabacaacaacacaeeccdbeebaddddeeececbababdebbadccdbaaedacadaccccceaebecedddebdcdddddeaebecabadecedddaedececddcdcabdcccedceaaececdddbdeddadabdceeaeaacceaedeeccaccedaeacccddabccdacadebcebdbbbbcedeacccdbdcbdcddebbceacbbbcedecabbecbcdcbddcebdebbeeedecbaeacddbcacbcebbbbceabdcceeaeaadeceddcabccbceeaaabebeaedaabecccaeaceacbeddddaaaceaaceceeabebcabededeeaeddedabeaeeaddaadccaeceabaeeadacaceedcbbcecededdecdbaadbecbaeccedaebacbebeecbadcadeadeebcbcbadbecdaebdbccaeaacdcdeadbdbaceacabcbcccbdbdbedeacbbdaebbbcccbeacddadcaeabcaecccbbecadcdddbcdbcbbcbadabaeccedbbcbdbecabcddeacbbeacdcdadecdccdecbdeebadceadbbdabbcedbceabadaaacdddeccbcbddeeceeebbedbcbabdbccdebaaacbbeccccebdbddbacbabbbacacdecbbbcdaacbdcddbabccdeaeeaedeebedbeacddcceecaaadeacdeeeaeaededeadddeddbbacaddbbdbdceceaedceedbaeeadbdbdbeedbadbdacbdecebecaaeebdbadecabbcdeeaccababdcbdaaebdcbbbeabdcbabdecbcbdbcceadabdeabdbbabadcebabeccbbdbabcaeebdaedeebcbacaddeabdbdcbdcbcaecacabbbeaebecbccdacdcdcceeccaeceaedacadecaeaeebbeaebdcbbdcecabbbacebcadbeedeccbaaeebdbeeadceadddeaceadddbbccadccbdbacbddccaceecadbeeddeeecaebbdbeecababcbccccbaedacadcedbbeedabbeaacdeccedeeedaadaddddcebbeeaeaeaecabdbbeaedaaacbecabaaadcdabadddbeaaedbdaaebacbbaaaeaebaedbccbcabaeabcecbeacabecaddeeeaceadcebddccbdcaadcaaacacdeacecbabdaaebdddcdedebacaeadbeecaaebaaacdcebccceddddeceaacaebddaebaeeebddbbabcccadbedcabcdbececacececbcbcebbccbcedbdcecadcbbeebccddaebeeeaddabbcbbbbaaccedcaccbadadcaadbebebaadeaceacbbcacacddcccaababceacdadcaceabcecadeddeacdaacebaabebaebeaceaabddeeccbabbdcacbdacdbdeeacaadddcacbababeeccbbaccccdacdedccdcabbddecacebedaccbbeeabedeadbcadeeaadabceccbeeedbebbbabdbacdadbeabaddceebddebdbabdbdadedcacbbbdbcbbdaacabcadbeddcacacdbcdcbbbbaedbdbcadbbaaedbddbeecabcebbebaedcdcaabdeeccdeeccaeeacabeaaacbdaeabeecaaedabbceebbbdaeeabecdadebcbcddbaaaedceeaeecbeddaaadacdccaacdbbdabaaebccdcbeaedcbeabebaaadadddaadeeededdbebbecaabbdcedebdddaadacbbbcebbccadcbaddccedcedbdebbdddccdcbbbaceaaedbecccceacbeddcebbdeacbeeecbcadcccdabebdbeecbcdaccaacdbdbdabadcabccedcbabeeecceccddcedaaddceaeedcdbebaadcdeaebabaaaedceebccbceaebeeeecbaecbacdcbedabeccaeaeaaddddaaedbebbbbbbabcadcccaabcacacecceeaecddbdabddadccddeaacdbcdceccbebddcabbcbadeccaaddeaebebedacccdceceebaaeceeeddedbaedbbcbaddeddadcbeadbeebdbdebddcacdccccbcceddaabebdcbcabcabaadadabdabadbebebcdcbaedbdeaeeadacacebbeccdeccdadeaacaeccbbbacdecdaebaceadbddcaedabceaebcaaeadaccdabedcedbabbcbbebbeecbaecdcdbadadcebddbdcbeeaccdbcdeeaabdebeaaeedbabcbdbeaaaeededddaedccedeaeecdddabcdededacaabdcccddccacddbdeceadeecbcdcdbadbcdeaeceeadedccceecdabcccccbecedbebcaaccceacbedebeaeaaecdddadeacebbaadcaadbadcbddeddbeedccdeceaebeedbdadbebdbdaceabeeaecebeaaeedabcbbadebbdeeebeaddadaeaacdeacddadcbdaabbddcdedadbebaabccdddcdeeaeecbebdaceadeebceebceddabcadcbcabbaccbcdcadcbeddddcdaddebebcbcdaaddceebcdeeebbccebecdaeeabdadcaadaeeddeeabeaecbccadbbedddbddadeeabdaebbceceabcdebeaeeebebacbdeaeecddddbaecaebeabbcdcdeecdcaeeaeccddbcbedaecebcdddecdbdccbdebeebcebeadaecbbdadbdebcebdadedcedabacdbdbcecebbaedeaceacdacaeaeceaddaceeedeacecaedcecdddaceeadbbdeaccbdbcddeddeebadebdbbedcdebdbcaeaedecdbcecbeceadbdebdadceeadaadcbbcabbeddeccdedcacaebdadeaddccccaedcbcdddbcbededdebcdaebcccaacedbbccdbdccacbdbbedcbcdbedbdbaccaedecdbceadacbdbbcbcdbbbecbcaaceabeccabaabbaddedacbcbbbabcadaabebcbcaeabccdbaaebaedeadcebedaaaabdbcbbdadbbdeadeecbdabcbeceedeeabdcabbaeaeabcebdeddbddaceacbaeabacdacbccbbaeacacbcaeebabadbebdadcacacbbcbabecabaeebbbbbadaabcbcbdaeabedccdebdadcccdabbeebdaeeadbbccddeeababbbbaebeeaaaabdeeeececcbebbacadddbbddbbcadcdaebdcdcacaabdaabcaeedebbbaebcccebccdbbaeeadaaebeeccdbebecebeedecdeaebbcadaccbaddaaabcccaecbdabbbcabbedddcdadebccbbadeabdbdcededbeedaebbdaccedaeaacaacccbdbbdcdbcdcaedbbaabbbecdaddbabababecbbaadecaebbcededbecdcaedbbcceccedcededadbaccdbabaebbacdacebaacbceeaaabcbddcecdaaaedcbdaadcdbcbbacecdbcabcadeaceedabadbbddbaacacceeabaabacbeddebbccbecabeadcecbdbddeaebeabedceababddecbceeeddbbdbbbbeeededaebeebeabbdadeedcaaebdaccecccdaccbcdeaccaabeeaaeeabdccccebacaacdbabdecaecbdecdecbcdccbeaccadaecaeadaaacdeeebbbebbcbbeceacebbbbbdddababcdddddeecebebddedececeaaebcdccbedeadebceeeadcdabebcaaeedeccdabdcbdbdeabaeccbeedcceaccdceabdadecceadcabaaeadceadbbdcaceececdadbdbaababdddedbbdcdabeececaebdeedabdaabcabcdccaebeebacacaedaeabcaaaccaeebbeceabaaaeacaddcceaddbddbcbbbcaedcdceacdebbbceaacacdbeecbcbbdcbaeedeeabcdcbabcccedeadccbbceabdbbeedddacbddcdcbaebebddcbaaebecdabbeaaaddbbbcbeeaddececbeebdaedcdacdcabdeedebaeedbedcecbbbdecddadebcdbaaebecadaeadceadeeaedbcddbedebaaebeeadabeeeadedaadcedcbbbacbadecdeebbedbbdaeaedbaaabbdbbeddaaedcdbdadeebcaeacbbaaeaeaadedcaadddccaaaddcbcbadeeeabbaeeacceaabadecbacbceeadeccaedbaaeeecddddbdabcceadddbaebabbbdbaddcecdaeaeecccbadbdaddceccbcbeaaecacdbabaeccebeebddaceabbcbcaaddbcccdacddcececbbbceccacbbdcaecbebaeaeebceadcacddbaeadadddddbeeccacabecaebaacbaaabbabbaccadcdaaeececdabbcccbaaecaeebddbcbdaccbbbeacaadbddebeddcbbdecaeddbcdbcbdeabdadbcaacebadbaeaedaedecdaccedcedabeabaedbcdeaebabddccabcbadbbdecbcecaaedadacdbcdebeabcdacdbebddcbccadecdbbbebcbeccbadbeeedbbdbdabbbbbbdecbcddadbcebedbdebbadcddebdabddbccedcebbcaeeacacdedccdcebdddaccbbbccabbbeebbdbebdbecedaaabdacddbdedcbabedeabaceccadeedaacddcbbccbcbdcdbabbcdbeeebaabeddcdaebadcecddaadaabecebdbdbbcacaedabadbcecbadcddcebceaebdddeadcbbedcdbedcacbeeadcddeecdbeebbcedbbecceaeeaebacbeeaddbcbbccbeabeabebeeccebcdaacedcabdacdcaabacedcbadcaaabaaaabaeddacacaaeaaabedccebdccacdedbcdadecddabdeadebcdedeabddaebbaaadacadbddeccdbebecacdadbbaadbaedccbeacbaabcbdccddabdbbbbebeacaebecbedebaeadabccdbbcecaeaebcaeeccaceecaaeabdbdbeeadeadcdbddcdaedbccbadaaccccabbcccdedbeeadcabedccddaadedcbcdddedaecbbcbbacbbeabdbebabdaacecbaedbaeacecbcedcbcabdededabeddbdcecbbcaebabdecacdececabcaecdebdaddadbbbaadbecaacbcbebbceccddeeecceecdbeecbadbbadedbbedcdbeceecceddebdabbbcbdceabbadacccbcddedbacadadcdccbbbccbaecbedecedabbbcacdcddcbeccadabebbaccadceadaececcdebcbbadebcbadacdedbdedededbeacaeeecacabbcaeadeacaceaeddaecdecddceccdcecaaecdccaccbbbcbbccabeeabedaebdabaabaadcbcdebabeccdedbbbcadcbcbdaebabdeebbcabacedddbcbdbdbaecaacdadebabaeadbececebeebcdddabbedaeebbecacbeeabddbdcecbeabdcabbdcbebacdabdbbeddecdbbbabacbbbeeccdadcaabbcddebeacaadbecabcaebcedaededeabbccdbadebadabcbcccadeeaacddbedbcbecbbebbeacaaccebecdaabbabcaddceebacaebceebecaaadeedaedaedccdebccbaeaeebacbabecaeaaecbdabaadcbbdabeeebdaabdbbbcaeedceaecbeedededcecebcdaadddeacecbececdecaeeeacebbbcbbabdbbdaecaaacbbdecdbabbaceaabcebcebeddacedecdedabbaadddbaebdcacbcdcdecdeaaeddeaaabbbdcccdccbcbabdcdeaaceecbaedaeeceddbebecdcbbdeeedeebdacbccddbdaecaedbbedaccbecdeaebbdcaceeaccabbadbbebeccceebbdaeccabacaeccbdbcbeebdeadbbeaadadabeabbcdcebaabbbdcdececccbcceaacbcbaeebdeaebcebaeaddabcdddaaddbcecaeaaedaceedaebadbddbadcbbbedecccbabdacbbacecaeebabdebbcdcdedcedadaacbbeeaddaeebcaccdcbdbccebdcaecbeddcbecceabcbdaddacdcddcabcbbabbddcbdddebcbeddbbabbedacceaedbabadcdaaabdcededceddedbccdcddeeeeadcaaccdeddeababcedaaedcdcaccbdbdabcaccceaabddccebadeababeaaccccaaabbbcdacabedbcbeadceecebccbacbdecdbdebbabeeddecdacdacbdaaeccdbeccbdbacbeabcaeedeceeddcedbeebeedacecddecabdabdbdabcadbdacdddcecccbcbcaecacdccecdecabbdadebdaabcacaedecbadbeebecebeacaddaabdeceeeddbdaddbaaebcaaadcaecedbcddcdbccbcddabebdcececdaeebcdccdbacbbaaecdadabdaacdcdbeadcbcecbbebbdbbadbdcebacaedebaeebebbddbccacecddcacebdceeacaeceebeeadcabacacbcbadcdeddecdeecbdacdccacdcceabcbdabeacaacaacbaecdcbadcbbeadebaabacdaecdbeacdacdaaeeedccadaadacdbaeabbedaeadbbbdbeeaeacccebececdbeeebbaeadcceedecebecabadaaeacadbdedbddddcabceccdbdaadceddebdbabeaeadbabbdaceabdbbeaecaecdccecaaceeadeebdeaebdaeaaaadcecdadcaabcdcdaaecbdaebabaeecbbcebeccdaedddcbcdedabeebccacedaddaacecdccabbacbabddccccdbaecaceecaabeeaecccaabbdcaaddcecbeceaaedbdccbecdabbbdabdababaaaacdadbdbebadcebebdbcbbcedecbacecedceaabceaaeedaeabecddaaccdaeaccddeddddebecdedbbcacbbeeababbdaceceaaeadddaddacdecdadbcedcebceeaacbceeeebdbdbcbdebcddabcdabedeebbbddbcadbabdedcdccbecbdceececbcddcbaeeddcceaacaebaddaedebbabcadbbceaeadcaaedaebbbcdedaaebbdbdadedbbbbdabcecdabbebaecbbddebeaecceaabebedecbddcceaecedbadaceccaadcdcbbabacaebecbaccbdadacdedaaacddcaaeeeeedabcbaaecceceeaabbcdedecedbcabbaadcadeabcccaddcecbbaebebceebedaebeddeeeaedabaadcdceddaaadebaabdadaeabbeadbbbaeeadaaadedeeaebbbeccbaeaddaacdbdacaababadccebbbbddcbceabbccccabbccaeaedcccbaeeabddaceabbbecbceaacabaaadaebdaeabcbceeebacdbeccdbaddbbcacbbeadbbddbdaecacbadbacdacacaaacbccebeadeadccbcccceebeeddebaaeabcbececddbccdaddbbbbbadaeaeebeeeccdaaacceaeaaaaabacbdccdccbbebcbdcbacadacebbacdebaedcdeeaddccedcdeabebddccedbabeeaacedcceccaacdbeabecccadcccbccacbacebcdedcedddbeddacddcbbdeaaceaeebbceadbceebcadbbebbbaecedeecaaebddaecebdeebaeeebcdcbebacedaaadaadbbdaedeaedbbbadbecdddbeaaabcccbcccecedaccaedbecbbcdcddebdcceccedacbabaaddbbdeaddccaebabadbaeeebaeadbeebaeeaaabbaaebeceaabeeaebcdbbaeadbbdbacdadbedecdeebeeeaebcbaaacdccacacecabbdbceddddeabeaaaaadaaadcebbaacedcabcdbbcacaccadbbdaebcdedadddccaaabaecebbbccabdabccabebeadebaaebdeaabbcebcbcdaadbaedcbadbcdbbbeccccaaddedbcddebaddbcaabbecccaecbaacaedabcaccaceacecacabdbacdeebaecdaeeebdddeadddeceabbbbacdaaaaeaabdaedbdcdcaadeecedeeddacedebcdebedecbecceecabcdeecbcbbadabedbadbeebbaadcebbbceacccdeabdaaeabdabebbdcbcdeaecdcbcbacdbceaccdeaeaedabdeaabcdcabbbbbbdaaccbabddcbcddbcaacedcdadaaadcbbbbddddeeeeaecddbcbbbdeaaecbcbacdeccecabdcaeaedcdcadbcacbdbeadaeebabacbceabacdcbecbacbdedabacaeddeaadaeaaebcccbdadcdccaedbaddcbaacadbadeeeebbdeccdeeeeabadedeebbcdbdbbecdbeeadeaeeedebcabacebedbceeacbcccbaadacbeeedaaeebbedeedddaebaaaeeaddcddaccbdadbdbbbadebaaecedebddeaedbdebdeddcabbcdcdceabdcdbbabdeddeedcbdbcbbbdcdbddadadeaebbdddaaeddceadcbaececdadcddcbcbebeccdbbbebabaddabeebabeceeededdabaaabacdcdbdaabceccdbbdaceaeedbaccddcaeaccbededecaccbcaebdabcdadbcbbdedcacbeccdacdaabccbabedbeaeaeaabeadeceddbbcdddbeaedcebcccececadaddeeeebbeebcbaeadeadbcddaaaccebdedbddbeeabdaccaaacadebbdacaaebbadeeabbcbbbcbcacaaaecacddbbcbdcdcebbedadcabcccdebabbecabcaebbdcaedbdaddeaecaeebbadceccadddacdeeabdeaeecceccbedbdabdceedcccadbaccaeadecebbecadeeebdaeaebedaeaeeebdddebbaebadacbcbaebebbcdccaeedacaccdeabecabaeccdebaeaeadbeebeacbcbdaaedeebcaddbbdccbdaebdeadabcdbcbaeacadaeaaddececcaabdcceaeeddebdcecedacedaaedacddbbcabdbeaabbddedaebeeeccceeebaecddbcecaecccecdaaddeeaedbdbacbabedbabeeccedcdbbacbabbacbdbaeeddadaabdbaecbdeaaddeacebababcacaecbebdabedeceabdbacdebeadcbeabedacdbedaceadcbeabddecdcccbbebaeebcbaadcdbbddbdccbdcceadaeaceaccecbcebabadcabbeaddcacecbcccbebaeaddbbeebaaaeebbeebdadcdaeebeacedcbeaaadedbcdbbdeeeedaebccacacedaacbebbcccbdbdbbadedbdbadddacabadeabeebeeadedeeadacccdcdeebbebbdeabbbecaadaebddedddeaceebbccdabeeeeaaeacdbecdbbabbcdaacddaeaaaaddcddbcdcdcdaebdadbdaedcdedaaddbadecccbddeeedddbbcadcaeedcbdabbbdbcbbbabaeeceacbdbeeadbeeabdbadbeabbbdaecbeeddbccbceaccbbdabbaedbecabbdadcabcebbbaeebddbcebceaadcadeaedabadbdcbeccedecbceddddaeaebabadadcdbddeecacdbebbdccdbeeaabedccceabecdeedcbedddceedbcccdddcbdebddddbdabaacdabaeedbabaddececeadadaeedddabdbcdcbdeddebbbaaeccdccbaccacbeddeebedeabccdacbaabadebcbdaeceaccaadedaacdbebecccbedcdddecddcaacbbceedadcbdbddaaebebdecbddeeeceaacccabcebdaecbbaeedbdbddddcbecbdddeebecdaacbbdcaaedbbaddedcdcabddbdcbaacaadaabddbceeecbdebaadbbccdecaedbedcbebcdacdcaceaedcebddacbcbaecebcddcbdecaabeaebeddaeddecdbcdeabbdaeedabdccbcadededccccbebcdeacdbdabdcdcbbdbdaabcdedbbdeeebcadeebecbbeedbdebcababdeeaeeecaccacaaeceeabaacbccadceeecccaceaacbcaccdebeaeccdebbdeebcadadcadbceecabdcebebabeaebbaadadadcbcbacdecbebebbaabddcdaeaedcbecaaecabcadcdeadddbadbaccdbbcdbebceceeaabcdbecbcdbadeedaccaecbaeecabeacabbacebceeebebaddeeccdceaacbaacddecadbebadbddbcbaeaeaaccdebadcacddedadeadbbcabccdcceeecbacbccbadaeaaacecebbadadbadeeaabaebcedbeeceaeccceebbdbecbcebceeaaeaebcdcabcaeddacaaacbdedbbaaeebbaabdbdacabecbddaedaddeeeedccecdecdbcccaeecddebbbadbeebcaabccdaaeacabbaddcaceabebedebbddaaaacddedcedcedcbbabaabbaeaeaddbcccdeddacadacacabeebcbabbbbedecccecebbebccadeddacbbacedaebdbcccdcbaccbceeabaedeeaccbaebacdbddaebeecdcaeccddaadedddcadedcbdcecbbaaabcbbeeebaabdeeebbedcdcabaeddcccedaedbddbdbcdadebecacbdaeabebeddabadcbbbedbebbededeccecabaddddedebdebeebccbabcedcddebcccbcbdaaebdcecbeedceededecaedacebaeedaecabeaedaeeadebdccceedbaddeebebddcbcedaacdcaebaccdeabeabcabbdcaecedcbadbcacbbabbdabaedaeaeecdacbeddbcdbadeacbcaacecaabcabcacdbaecbbaebaaabdcedcaabdcebbdedcabdbaaeecbeebddeceddacebadececacccdebbcdecdceadcebacbaabdddeccbedeeaedaeabadbcacaedaacccacaabebcaebeccdabbceddeccdbbeeeccddbaabbaeecddabcbdbeadacaccbbbabeceedaabcbbbaaaedadebaaceabcdcddabeaecaebbaceceeabbebdeaecbccdbcadceebeacaeceaebaecacccdebcabecbecdbaacbeccbcacebdaaceedddedbbbdcdbccddaececadbbbcbddbdbadeccdebacbebababbdedcaceeaeebcdaabdbacccdeabadcdebbaaeaadaebdcaeedacedcaebcaeabdcdababeacbaddaeccceedcaedddbcecacadbacadbcdaeeceeeabbbadcaaaadaebdccabbbadedadbdcdcbccdabeedecadabdedaaacaacdaecbcdbbcaabdcdbebcaccbeecbebcebaadccddebdbdceedccbcbdcecebeabceecbcaddaccaceaccebbdeecddbecdbbadeaedaecceacceeddcacccdeeeecdbbbccaaccecaecaabcccdabdbbaabebdcecbbabaeadddeebacabebceabbdeabbdcabdaebdcecceabaabcbbbadbeeabacebaaeddedaecaccddebebccdbdcabdecbdaeababbcdeccadddbecbebbedecaaecbdbccadbddddaaacecebbbbcbedadbbabcbdccccedcdbdbaaeecacbbecaaccadcdaebdcbededceeedddedabadeadadbcbdceaebcddddebcccdeebbacecaaccadedcbadabdbedcbbebdccbbecacabbdbacdbdecdebcacacdabddbdbbecbcecddcceabebebeeabcdceecedbeddacbcaedaeedabcbdeedabaeecabacddeccdebacdbdeabcebcabeddbdcaebacdcbcbbbbeaeceebadecebaddcdeddccebabdccaadaeadbdadeddcaecacdeaccbdabdcbdbdcdadceeedaecbeeebaaedeaddbdbeabceecbccbebeaceeeaceabddbdbbedabbbdaecaddabdcbddadeccbaeeaecdeaeedaecaacbbaaecabccbcbadeabdcabedbcecbbdedaebedcdbeebedebdcedbbdceeedbbcaabadabcdcabaebacdcbcdcccaceeceddadeaadebceebccbacedbdaceadeacadeecceeaebadceeacaacdedaecaeccdcaaabceeaabdecaabceeaacaadaaddccdbdbadcebeddbbadadbbaaadeadbbbdddebebbbaabcdebbcaeddbbacabdddedeababbabdadeecdabbeecdaeaedcdddcdedebceacaccadcccabadbedaecbeaeebeceacbbbabbcdeadeebaabeeccadddbccaeeddddbbeecaecbdccbeadacdbaedadaeddcaeaedaeaeacdadcedcdabddacededcbeebaabeadaabbeeddeeababbaacabdbeadeadeedceadcbecddacaeaceeebcecadaebdccaeacdedbbecacabddeeeadecddbacddbebeedacdeddacbbbcebbaecaecadccccdebccdcdcbacbcccbdabdadaababdacebddcceddbdecdacbadaaaeaeddeeabcebbccbdbbebedabbdbdcaedebaaecddbacbaaaccdadeddebebdbabeceeaadadaabceecdceadebecacacbaebaebebceccdeebcabedaabcabccabedbbebabaababedeebaabcdaacdcccebcaecceedeebeabeadacbbcdcbbcdbddbcdadbcaceddedabeebbeedaabbccacdacedabbdeacbeebceadcaeacdbceaceaaaeebcbbebdeddbbbeddcbcabaacaadebcdceadcadaececbbbeecddcbbdbcdecaaacedbcbdbeabbbbdeceaebdcddbbebdbebbbcabccaecacbbcaeadcdbcdaadccaceddedaabaabcacacaeddcceadbabdbcbdaeeacabacdcdcbccaedeceacaaadabeabbceeedaabdcdbddebddadbadbbddedecabbbbdabecedbbcceacaacbebdeddbedbeabdbbadbeacbeabceaddadeeececeaaacecaaacbdcdcbcddcecadbbeceadebadececacaeaeccbdbddeadabcebaeeaeeabeccebdaeadaeedddaeaaedabcdbcaaeacabdcaaccbadbeabdbcbdcbeddabdacebcecccddcacbbccedbecbdcdbdeeaaaeeecacbacabbcccccaecacdececaeceacacecbdcbceaedcbdabeecbabbeaadbaaaeceabbaacbbbbddaeeadabaadcadcbeeabececbededccbddcbcdeacaacbabaaaebdcbcecdcbdbbdcbaeeebdaaecccbbbabdaddbdceaaacabdaccaeaccecdadcebdbceddbcbbcebdbbaccedbbeeceaeadbbbaabdaeaeccdcebceebdddaaaaeaeabdbebeddbadbbbddadadadcbccebecccdeeeaaeddaacadbbbbccecbbcdcebeeeaecbedbeecabcdceaeeadaeccecbbcddbacbdbcbbdceeedccbdeadbeeaacaeccaebdabbecabaeecdbdbcacdbcbadcbeeedbcebaacdaccbbcecdbdebabacddddaeadcdddcbedabbcbedcadbabeedeecbadbacbddaccadebcbaadbcbdabedccbbecbbbccabdaeacdaddcdceebedccdaddcdeccacecacbcedaccdbdbcbeeedaabbeddcaaacaceabcecacebcdacecaceabbccccdcbacaeaeebbbbecbecdbcabecccdebacbadcbaeecdcaaabaaecaacdaaccdbaaacdadecadaeecccdacebdddecbaacedbedbcbbbeabdabcdcedadcaebbebaeeebddbebcddabbbdedbeadeaebaccabecdbcbeaaabadbeccccaddbacddacbaeaecccadedaddedbebcecebcdadaccdcbedacbcbdcddceedaebedecdedeaecbeadbccedcbbaedccbeaeebaababeccdbdbdbcddcaaebcbcbaabbdaaabcccbbaeeaabbccddeebadebbdcebdaadbabdcacdeeeacedeaccabcbbeaabcdaecbdacecdabdbabaaadcadcdacaeeccabcaebcdcceaeccbaddecbceccddbbdbbbbadbdadeeaedcdbeddbbbddcebeaadbedebdddbcbddaacaceccbacbacebaebbbcbacddbaacedeeedbcadaaebdababbddabbcbddaccdcddbbdeabaacaeaaddacedaaceeedcedcecceaadbdebdcbeeaabaeca".to_string();
    let res = "ddbcddbabee".to_string();
    assert_eq!(Solution::longest_dup_substring(s), res);
}

// Accepted solution for LeetCode #1044: Longest Duplicate Substring
// Auto-generated TypeScript example from java.
function exampleSolution(): void {
}
// Reference (java):
// // Accepted solution for LeetCode #1044: Longest Duplicate Substring
// class Solution {
//     private long[] p;
//     private long[] h;
// 
//     public String longestDupSubstring(String s) {
//         int base = 131;
//         int n = s.length();
//         p = new long[n + 10];
//         h = new long[n + 10];
//         p[0] = 1;
//         for (int i = 0; i < n; ++i) {
//             p[i + 1] = p[i] * base;
//             h[i + 1] = h[i] * base + s.charAt(i);
//         }
//         String ans = "";
//         int left = 0, right = n;
//         while (left < right) {
//             int mid = (left + right + 1) >> 1;
//             String t = check(s, mid);
//             if (t.length() > 0) {
//                 left = mid;
//                 ans = t;
//             } else {
//                 right = mid - 1;
//             }
//         }
//         return ans;
//     }
// 
//     private String check(String s, int len) {
//         int n = s.length();
//         Set<Long> vis = new HashSet<>();
//         for (int i = 1; i + len - 1 <= n; ++i) {
//             int j = i + len - 1;
//             long t = h[j] - h[i - 1] * p[j - i + 1];
//             if (vis.contains(t)) {
//                 return s.substring(i - 1, j);
//             }
//             vis.add(t);
//         }
//         return "";
//     }
// }

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(log n)

Space

O(1)

Approach Breakdown

LINEAR SCAN

O(n) time

O(1) space

Check every element from left to right until we find the target or exhaust the array. Each comparison is O(1), and we may visit all n elements, giving O(n). No extra space needed.

BINARY SEARCH

O(log n) time

O(1) space

Each comparison eliminates half the remaining search space. After k comparisons, the space is n/2ᵏ. We stop when the space is 1, so k = log₂ n. No extra memory needed — just two pointers (lo, hi).

Shortcut: Halving the input each step → O(log n). Works on any monotonic condition, not just sorted arrays.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Boundary update without `+1` / `-1`

Wrong move: Setting `lo = mid` or `hi = mid` can stall and create an infinite loop.

Usually fails on: Two-element ranges never converge.

Fix: Use `lo = mid + 1` or `hi = mid - 1` where appropriate.

Shrinking the window only once

Wrong move: Using `if` instead of `while` leaves the window invalid for multiple iterations.

Usually fails on: Over-limit windows stay invalid and produce wrong lengths/counts.

Fix: Shrink in a `while` loop until the invariant is valid again.