Week 4: Hash Tables & Sets

1. Overview

Welcome to Week 4! This week concludes our first phase on Core Data Structures. We are diving into Hash Tables (specifically HashMap and HashSet in Java). This is arguably the most important data structure for coding interviews — it appears in roughly 40–50% of all LeetCode medium problems, either as the primary technique or as a key supporting structure.

Hash tables allow us to trade space for time, achieving lightning-fast $O(1)$ lookups and inserts. They are the backbone of caching, frequency counting, and relational mapping.

Why Does This Matter?

Without a hash table, finding whether a number exists in an unsorted list takes $O(N)$ — you scan every element. With a hash table, it takes $O(1)$ — regardless of whether the list has 10 or 10 million elements. This trade-off (spend memory, save time) is the single most common optimization in software engineering.

Goals for this week:

• Understand how Hash Functions map arbitrary keys to array indices.
• Learn how Java handles hash collisions under the hood.
• Master the equals() and hashCode() contract for custom objects.
• Recognize patterns where $O(N^2)$ brute-force solutions can be reduced to $O(N)$ using a HashMap.
• Build intuition for when to use a HashMap vs. a HashSet vs. a LinkedHashMap.

Knowledge You Need Before Starting

• Strong grasp of arrays/strings plus two-pointer/window fundamentals.
• Basic object identity vs equality understanding in Java.
• Comfort with Big-O average vs worst-case complexity discussion.
• Ability to model problems as membership, frequency, and key-value lookup.

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2. The Core Mental Model: How Does a Hash Table Actually Work?

Hash Table Collision Resolution (Separate Chaining vs Linear Probing)

Select Key to Insert / Lookup:

Collision Resolution Strategy: Separate Chaining (Linked Nodes / Red-Black Tree)

Colliding entries form a linked bucket (or TreeBin when count >= 8 in Java HashMap). Load factor threshold = 0.75.

Before memorizing templates, you need to understand what is happening under the hood. This will help you reason about edge cases and explain your choices in interviews.

2.1 The Bucket Array

A hash table is, at its core, just a regular array with a clever trick for finding things fast.

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5.2 Keyword Trigger Table

Problem Keywords	Technique	Key Data Structure
"two numbers that sum to target" (unsorted)	Complement lookup	HashMap<value, index>
"contains duplicate" / "seen before"	Membership tracking	HashSet
"group anagrams" / "group by property"	Canonical key grouping	HashMap<key, List>
"first unique character" / "first non-repeating"	Frequency map + scan	HashMap or LinkedHashMap
"count occurrences" / "frequency"	Frequency counting	HashMap<element, count>
"subarray sum equals k" / "number of subarrays"	Prefix sum + complement	HashMap<prefixSum, count>
"longest consecutive sequence"	Set membership check	HashSet
"valid sudoku" / "valid board"	Multi-key existence check	HashSet per row/col/box
"top K frequent elements"	Freq map + heap/bucket sort	HashMap + PriorityQueue
"design LRU cache"	O(1) access + O(1) eviction	LinkedHashMap or HashMap + DLL
"isomorphic strings" / "word pattern"	Bidirectional mapping	Two HashMaps

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5.3 Common Traps & How to Avoid Them

Trap 1: Using a custom object as a key without overriding hashCode()

// ❌ Point only overrides equals()
Map<Point, String> map = new HashMap<>();
map.put(new Point(1,2), "A");
map.get(new Point(1,2));  // → null! Different hash → different bucket

// ✅ Always override BOTH equals() and hashCode()
@Override public int hashCode() { return Objects.hash(x, y); }

Trap 2: Checking containsValue() in a hot loop

// ❌ containsValue is O(N) — scanning all values
if (map.containsValue(target)) { ... }  // Inside a loop → O(N²) total!

// ✅ Maintain a reverse map if you need value lookups
Map<Integer, String> reverseMap = new HashMap<>();
// Or restructure to use keys instead

Trap 3: Modifying a map while iterating over it

// ❌ ConcurrentModificationException!
for (String key : map.keySet()) {
    if (shouldRemove(key)) {
        map.remove(key);   // Boom!
    }
}

// ✅ Use iterator.remove() or collect keys first
Iterator<String> it = map.keySet().iterator();
while (it.hasNext()) {
    if (shouldRemove(it.next())) it.remove();  // Safe
}

// ✅ Or collect removals separately
List<String> toRemove = new ArrayList<>();
for (String key : map.keySet()) {
    if (shouldRemove(key)) toRemove.add(key);
}
toRemove.forEach(map::remove);

Trap 4: Forgetting putIfAbsent(0, 1) in prefix sum problems

// ❌ Missing the base case
Map<Integer, Integer> prefixCount = new HashMap<>();
// If the subarray starting at index 0 sums to k, we'll miss it!

// ✅ Always initialize with the empty prefix
prefixCount.put(0, 1);  // "A prefix sum of 0 has been seen once"

Trap 5: Integer unboxing NullPointerException with map.get()

// ❌ map.get() returns Integer (boxed), not int
int val = map.get("key");  // NullPointerException if key doesn't exist!

// ✅ Always use getOrDefault for primitives
int val = map.getOrDefault("key", 0);

Trap 6: Isomorphic string mapping in only one direction

// Problem: "paper" → "title" (p→t, a→i, e→l, r→e) — valid
// But also: "foo" → "bar" (f→b, o→a, o→r) — INVALID, 'o' maps to both 'a' and 'r'
// AND: "ab" → "aa" (a→a, b→a) — INVALID, both 'a' and 'b' map to 'a'

// ❌ Only mapping s→t direction misses the second case
// ✅ Use two maps: one for s→t, one for t→s
Map<Character, Character> sToT = new HashMap<>();
Map<Character, Character> tToS = new HashMap<>();
// Check both mappings are consistent

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6. Worked Examples (Step-by-Step Walkthroughs)

Example 1: LeetCode 1 — Two Sum

Problem: Return indices of two numbers that add up to target.

Thought process:

1. Brute force: check every pair → $O(N^2)$.
2. Key insight: for each nums[i], we need target - nums[i]. If we've already seen that value, we're done.
3. Store (value → index) in a HashMap as we scan.
4. Check before inserting to avoid using the same index twice.

nums = [2, 7, 11, 15], target = 9

i=0: num=2, complement=7
  seen={} → 7 not found → add {2:0}

i=1: num=7, complement=2
  seen={2:0} → 2 FOUND at index 0 → return [0, 1] ✅

class Solution {
    public int[] twoSum(int[] nums, int target) {
        Map<Integer, Integer> seen = new HashMap<>();  // value → index

        for (int i = 0; i < nums.length; i++) {
            int complement = target - nums[i];

            if (seen.containsKey(complement)) {
                return new int[]{seen.get(complement), i};
            }

            seen.put(nums[i], i);  // Add AFTER check
        }

        return new int[]{};
    }
}

Complexity: Time $O(N)$, Space $O(N)$. Each element visited once; map stores at most N entries.

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Example 2: LeetCode 49 — Group Anagrams

Problem: Group strings that are anagrams of each other.

Thought process:

1. Two strings are anagrams if they have the same characters in the same frequencies.
2. We need a canonical key that is identical for all anagrams.
3. Option A: Sort each string → "eat" and "tea" both become "aet". Cost: $O(M \log M)$ per string where M = string length.
4. Option B: Build a 26-character frequency array → "eat" becomes [1,0,0,0,1,0,...,1,0,0] (counts for a,e,t). Cost: $O(M)$ per string. Better.

strs = ["eat","tea","tan","ate","nat","bat"]

"eat" → freq array → key="#1#0#0#0#1#...#1#..." → group A
"tea" → same freq   → key="#1#0#0#0#1#...#1#..." → group A ← SAME KEY
"ate" → same freq   → key="#1#0#0#0#1#...#1#..." → group A ← SAME KEY
"tan" → freq(a=1,n=1,t=1) → different key → group B
"nat" → same as tan → group B
"bat" → freq(a=1,b=1,t=1) → group C

Result: [["eat","tea","ate"], ["tan","nat"], ["bat"]]

class Solution {
    public List<List<String>> groupAnagrams(String[] strs) {
        Map<String, List<String>> map = new HashMap<>();

        for (String s : strs) {
            // Build frequency key — O(M) instead of O(M log M) sorting
            int[] freq = new int[26];
            for (char c : s.toCharArray()) {
                freq[c - 'a']++;
            }
            // Convert array to string key: "#1#0#0#0#1#..."
            // Using Arrays.toString() also works but is slower due to brackets and commas
            StringBuilder sb = new StringBuilder();
            for (int f : freq) {
                sb.append('#').append(f);
            }
            String key = sb.toString();

            map.computeIfAbsent(key, k -> new ArrayList<>()).add(s);
        }

        return new ArrayList<>(map.values());
    }
}

Why prefix # before each frequency? Without it, frequencies [1,12] and [11,2] would both produce key "112" — a collision in our key design! With # as separator: "#1#12" vs "#11#2" → unambiguous.

Complexity: Time $O(N \times M)$, Space $O(N \times M)$ where N = number of strings, M = max string length.

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Example 3: LeetCode 128 — Longest Consecutive Sequence

Problem: Given an unsorted array, find the length of the longest sequence of consecutive integers. Must run in $O(N)$.

Thought process:

1. Sort and scan? That's $O(N \log N)$ — too slow.
2. Key insight: put everything in a HashSet. Then for each number, check if it's the start of a sequence (i.e., num - 1 is NOT in the set).
3. If it's a sequence start, count upward: how many consecutive numbers follow?
4. Each number is visited at most twice (once for the "is it a start?" check, once for the count) → $O(N)$ total.

nums = [100, 4, 200, 1, 3, 2]
set  = {100, 4, 200, 1, 3, 2}

Check 100: is 99 in set? No → start of sequence. Count: 100, 101? No → length=1
Check 4:   is 3 in set? Yes → NOT a start, skip
Check 200: is 199 in set? No → start. Count: 200, 201? No → length=1
Check 1:   is 0 in set? No → start of sequence. Count: 1,2,3,4 → length=4 ✅
Check 3:   is 2 in set? Yes → NOT a start, skip
Check 2:   is 1 in set? Yes → NOT a start, skip

Answer: 4

class Solution {
    public int longestConsecutive(int[] nums) {
        Set<Integer> numSet = new HashSet<>();
        for (int num : nums) numSet.add(num);

        int longestStreak = 0;

        for (int num : numSet) {
            // Only start counting from the beginning of a sequence
            if (!numSet.contains(num - 1)) {
                int currentNum = num;
                int currentStreak = 1;

                while (numSet.contains(currentNum + 1)) {
                    currentNum++;
                    currentStreak++;
                }

                longestStreak = Math.max(longestStreak, currentStreak);
            }
        }

        return longestStreak;
    }
}

Why iterate over numSet not nums? Iterating over the set avoids processing duplicates. If nums = [1,1,1,2], iterating over nums would check 1 as a sequence start three times — wasted work.

Complexity: Time $O(N)$, Space $O(N)$. Each element is a sequence start at most once, and each element is visited in the while loop at most once.

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Example 4: LeetCode 560 — Subarray Sum Equals K

Problem: Count the number of subarrays that sum to k.

Thought process:

1. Brute force: check all subarrays → $O(N^2)$.
2. Key insight: use prefix sums. If prefixSum[j] - prefixSum[i] == k, then the subarray from index i+1 to j sums to k.
3. Rearranging: prefixSum[i] == prefixSum[j] - k.
4. So for each j, we need to know how many previous prefix sums equal prefixSum[j] - k. Store prefix sum frequencies in a HashMap.

nums = [1, 1, 1],  k = 2

prefixCount = {0: 1}   ← empty prefix initialized
runningSum = 0

i=0, num=1: runningSum=1, needed=1-2=-1 → count+=0. add {1:1}
i=1, num=1: runningSum=2, needed=2-2=0 → count+=1 (found {0:1}). add {2:1}
i=2, num=1: runningSum=3, needed=3-2=1 → count+=1 (found {1:1}). add {3:1}

Answer: 2  ([1,1] at indices 0-1 and [1,1] at indices 1-2)

class Solution {
    public int subarraySum(int[] nums, int k) {
        Map<Integer, Integer> prefixCount = new HashMap<>();
        prefixCount.put(0, 1);   // Critical: empty prefix
        int count = 0, runningSum = 0;

        for (int num : nums) {
            runningSum += num;
            count += prefixCount.getOrDefault(runningSum - k, 0);
            prefixCount.put(runningSum, prefixCount.getOrDefault(runningSum, 0) + 1);
        }

        return count;
    }
}

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7. Problem-Solving Framework (Use This in Interviews)

Step 1 — Recognize the Pattern (30 seconds)

Ask yourself:

• Do I need to check if something exists? → HashSet
• Do I need to check what value is associated with a key? → HashMap
• Am I counting frequencies? → HashMap<element, count>
• Am I pairing elements (A + B = target)? → HashMap<value, index/count>
• Am I grouping things by a shared property? → HashMap<key, List>
• Is this a subarray sum problem? → Prefix sum + HashMap

Step 2 — State the Brute Force (1 minute)

Always lead with correctness:

"The naive approach is $O(N^2)$ — check every pair. It gives the right answer but won't scale."

Step 3 — Identify the Bottleneck

"The bottleneck is the repeated inner loop lookup. Each time we scan the array for a complement, we're doing $O(N)$ work. If we could make that lookup $O(1)$, the whole solution becomes $O(N)$."

Step 4 — Apply the HashMap Optimization

"I'll use a HashMap to store elements we've seen so far. For each new element, I calculate what I'm looking for and check the map in $O(1)$."

Step 5 — Handle Edge Cases Out Loud

Always mention these:

• Empty array/string
• All duplicate elements
• Negative numbers (prefix sums can be negative!)
• Target = 0 (complement of every element is itself)
• k = 0 in subarray sum (empty subarrays — does prefixCount.put(0, 1) handle this?)

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8. 7-Day Practice Plan (21 Problems)

Day 1: HashSet Basics

1. Contains Duplicate (LC 217) — Simplest possible HashSet problem
2. Intersection of Two Arrays (LC 349) — Set intersection
3. Happy Number (LC 202) — Cycle detection with a set

Day 1 Focus: For LC 202, the key insight is that if a number enters a cycle, we'll see a repeated value. A HashSet detects cycles by checking if the next value was already visited. Ask: "Why is a HashSet better than a list here?"

Day 2: HashMap Fundamentals 4. Two Sum (LC 1) — Classic complement lookup 5. Isomorphic Strings (LC 205) — Bidirectional mapping — use two maps! 6. Word Pattern (LC 290) — Same as isomorphic strings but with words

Day 2 Focus: LC 205 and LC 290 share the exact same pattern. After solving one, solve the other in under 5 minutes. Notice what's reused.

Day 3: Frequency Counting Patterns 7. Valid Anagram (LC 242) — 26-element frequency array vs. HashMap — know both 8. Ransom Note (LC 383) — Same pattern as anagram, different story 9. First Unique Character in a String (LC 387) — Two passes: first to count, second to find

Day 3 Focus: For LC 387, can you do it in one pass using a LinkedHashMap? (Insert characters in order, then find first with count 1.)

Day 4: Grouping & Matrix Mapping 10. Group Anagrams (LC 49) — Canonical key design 11. Valid Sudoku (LC 36) — Three HashSets per position: row, column, box 12. Top K Frequent Elements (LC 347) — Freq map + bucket sort (avoid heap for O(N))

Day 4 Focus: LC 347 has two approaches. The heap approach is $O(N \log K)$. The bucket sort approach is $O(N)$. Know both and when to use each.

Day 5: Advanced Search & Subarrays 13. Longest Consecutive Sequence (LC 128) — The "only start from sequence beginnings" trick 14. Subarray Sum Equals K (LC 560) — Prefix sum + HashMap — master this pattern 15. Contiguous Array (LC 525) — Reframe 0s as -1s, then it becomes subarray sum = 0

Day 5 Focus: LC 525 is a clever disguise of LC 560. After solving 560, analyze 525 and identify the transformation. This is a senior-level pattern recognition skill.

Day 6: Design Challenges 16. Design HashMap (LC 706) — Implement from scratch: array of lists 17. Design HashSet (LC 705) — Same, but without values 18. Insert Delete GetRandom O(1) (LC 380) — HashMap + ArrayList working together

Day 6 Focus: LC 380 is a common system design warm-up problem. The trick: the ArrayList enables O(1) random access; the HashMap enables O(1) deletion by swapping with the last element.

Day 7: Complex Implementations & Review 19. Find All Numbers Disappeared in an Array (LC 448) — Solve twice: once with O(N) space, once in-place O(1) 20. Minimum Window Substring (LC 76) — Hard: Sliding window + two frequency maps 21. Longest Palindrome (LC 409) — Frequency map + parity check

Day 7 Focus: LC 76 is a hard problem that combines Week 2's sliding window with this week's frequency maps. Don't be discouraged if it takes time. Walk through the two-map comparison approach carefully.

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9. Mock Interview Module

Problem: The Fraudulent Transaction Detector

Context: You are building a risk-analysis engine for a fintech company. You are given a real-time stream of Transaction objects. A user's account is flagged if they make two transactions within K seconds of each other.

class Transaction {
    int id;
    String accountId;
    int timestamp;
}

Question: public boolean hasFraudulentActivity(Transaction[] txs, int k)

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Step 1: Clarifying Questions & Expected Answers

• Candidate: "Is the array sorted by timestamp?" → Interviewer: Roughly chronological, but don't rely on perfect sorting.
• Candidate: "Can an account have more than two transactions?" → Interviewer: Yes. Any two within K seconds triggers the flag.
• Candidate: "Should 'within K seconds' be strictly less than K, or less than or equal?" → Interviewer: Less than or equal (<= k).
• Candidate: "What should I return if txs is null or empty?" → Interviewer: Return false.

Tip: The <= vs < clarification is subtle but can change output on boundary test cases. Always ask.

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Step 2: Brute Force

// Time: O(N²), Space: O(1)
public boolean hasFraudulentActivity(Transaction[] txs, int k) {
    for (int i = 0; i < txs.length; i++) {
        for (int j = i + 1; j < txs.length; j++) {
            if (txs[i].accountId.equals(txs[j].accountId) &&
                Math.abs(txs[i].timestamp - txs[j].timestamp) <= k) {
                return true;
            }
        }
    }
    return false;
}

Interviewer: "Millions of transactions per day. This crashes our servers. Single-pass solution?"

How to spot the optimization:

• We're repeatedly scanning all previous transactions for the same accountId.
• If we stored the most recent timestamp per accountId, we'd only need the last entry — no scanning needed.

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Step 3: Optimized Solution (HashMap)

// Time: O(N), Space: O(U) — U = number of unique accounts
public boolean hasFraudulentActivity(Transaction[] txs, int k) {
    if (txs == null || txs.length == 0) return false;

    Map<String, Integer> lastSeen = new HashMap<>();  // accountId → most recent timestamp

    for (Transaction tx : txs) {
        if (lastSeen.containsKey(tx.accountId)) {
            int previousTime = lastSeen.get(tx.accountId);
            if (tx.timestamp - previousTime <= k) {
                return true;
            }
        }
        // Always update to the most recent timestamp for this account
        lastSeen.put(tx.accountId, tx.timestamp);
    }

    return false;
}

Walk through this aloud:

"I maintain a HashMap from accountId to the most recent timestamp. For each new transaction, I check if we've seen this account before and if the time gap is within K seconds. This makes the check O(1) per transaction, giving O(N) overall."

Why only store the most recent timestamp?

"If the array is roughly chronological, the most recent timestamp is the only one that could form a valid pair with the current transaction. An older transaction would have a larger time gap — not smaller."

Important caveat: If the array is not sorted at all, this approach can fail. Discuss with the interviewer:

Scenario (unsorted): txs = [{acc:A, time:100}, {acc:A, time:5}, {acc:A, time:102}], k=5
With lastSeen approach:
  time=100 → lastSeen={A:100}
  time=5   → 100-5=95 > 5, no fraud → lastSeen={A:5}    ← replaced 100 with 5!
  time=102 → 102-5=97 > 5, no fraud → returns false

But {time:100} and {time:102} differ by only 2 seconds → should return true! ❌

Fix for unsorted input: Store ALL timestamps per account in a sorted structure.
Map<String, TreeSet<Integer>> lastSeen → for each tx, check floor/ceiling in O(log N).

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Step 4: Follow-up Questions

Follow-up 1 (Distributed Systems): Interviewer: "The system is distributed across 50 microservices. A Java HashMap only lives in one JVM's RAM. How do you redesign this?"

Expected thought process:

• Replace the in-memory HashMap with Redis (an external distributed key-value store).
• Each service does: GET <accountId> → compare timestamps → if fraud, flag → SET <accountId> <timestamp>.
• TTL optimization: Set a TTL of K seconds on each Redis key. If the key expires naturally, any new transaction for that account is automatically safe relative to it. This keeps memory lean without manual cleanup.
• Atomicity concern: What if two transactions for the same account arrive simultaneously at different servers? You need Redis atomic operations (SET NX, EVAL Lua scripts, or Redis Streams) to avoid a race condition.

Follow-up 2 (Storing more transactions): Interviewer: "What if we need to track ALL recent transactions per account, not just the most recent, to detect complex fraud patterns (e.g., 5 transactions in 10 minutes)?"

Expected thought process:

• Change the map to Map<String, Deque<Integer>> — a sliding window per account.
• For each new transaction, add the timestamp to the deque. Then evict timestamps older than K seconds from the front.
• If deque.size() >= threshold, trigger fraud.
• This is essentially a sliding window per account — combining Week 2 and Week 4 concepts.

Follow-up 3 (Space optimization): Interviewer: "What if there are 100 million unique accounts? Memory is a concern."

Expected thought process:

• For this scale, even $O(U)$ space (one entry per unique account) may be impractical in a single machine.
• Solutions: (1) Sharding — partition accounts across machines by hash of accountId. (2) Bloom filters — probabilistic membership check with minimal memory. (3) Time-bucketing — only keep accounts active in the last K seconds using a time-indexed structure.

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10. Connecting to Other Weeks

Hash Tables rarely appear alone in real problems. Here's how this week connects to the roadmap:

Week 1 (Arrays) + Week 4 (HashMap):
  → Prefix Sum + HashMap = Subarray Sum problems (LC 560, LC 525)
  → This is one of the most frequent interview patterns!

Week 2 (Sliding Window) + Week 4 (HashMap):
  → Variable Sliding Window with character frequencies
  → Minimum Window Substring (LC 76), Longest Substring without Repeating (LC 3)
  → The window uses a HashMap to track what's inside

Week 4 (HashMap) + Later weeks (Graphs):
  → Adjacency list representation: Map<Node, List<Node>>
  → Visited tracking: Set<Node> visited
  → BFS/DFS become O(V+E) partly because HashSet lookups are O(1)

Week 4 (HashMap) + Later weeks (Design):
  → LRU Cache = HashMap + Doubly Linked List
  → The HashMap gives O(1) access, the DLL gives O(1) eviction

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11. Quick Reference Cheat Sheet

╔══════════════════════════════════════════════════════════════╗
║              HASH TABLES & SETS CHEAT SHEET                 ║
╠══════════════════════════════════════════════════════════════╣
║ HASHSET — Use for membership / uniqueness                    ║
║  add(x), contains(x), remove(x) → all O(1) avg             ║
║  Patterns: seen-before, dedup, cycle detection              ║
╠══════════════════════════════════════════════════════════════╣
║ HASHMAP — Use for key→value mapping                         ║
║  put(k,v), get(k), getOrDefault(k,def) → all O(1) avg      ║
║  Patterns: freq count, complement lookup, grouping          ║
╠══════════════════════════════════════════════════════════════╣
║ FREQUENCY COUNTING TEMPLATE                                  ║
║  map.put(x, map.getOrDefault(x, 0) + 1)                    ║
╠══════════════════════════════════════════════════════════════╣
║ COMPLEMENT LOOKUP TEMPLATE                                   ║
║  if (seen.containsKey(target - num)) → found!               ║
║  seen.put(num, i)  ← add AFTER check                       ║
╠══════════════════════════════════════════════════════════════╣
║ PREFIX SUM + MAP TEMPLATE                                    ║
║  prefixCount.put(0, 1)  ← ALWAYS initialize                ║
║  count += prefixCount.getOrDefault(runningSum - k, 0)       ║
╠══════════════════════════════════════════════════════════════╣
║ THE CONTRACT (custom keys)                                   ║
║  equals() == true  →  hashCode() must be equal             ║
║  Always override BOTH or neither                            ║
╠══════════════════════════════════════════════════════════════╣
║ COMPLEXITY                                                   ║
║  Average: O(1) per operation                                ║
║  Worst case: O(N) if all keys collide (rare in practice)    ║
║  Java 8+: buckets > 8 entries → Red-Black Tree → O(log N) ║
╚══════════════════════════════════════════════════════════════╝

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12. What's Coming Next

Week 5 introduces Stacks & Queues, which build directly on this week:

• Monotonic Stack problems often use a HashMap to store index mappings.
• BFS (which uses a Queue) relies on a HashSet to track visited nodes.

Week 6 introduces Trees, where HashMaps appear again:

• Storing parent pointers: Map<TreeNode, TreeNode>
• Memoization in tree recursion: Map<state, result>
• Serialization/deserialization of trees uses HashMaps for level-order tracking.

The pattern you will notice: HashMaps are the glue that makes other data structures O(1)-efficient. Almost every advanced data structure problem uses a HashMap internally. Mastering HashMaps now will make future weeks much easier.