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Yudong Jin
102 lines
4.7 KiB
Markdown
102 lines
4.7 KiB
Markdown
# Exercises
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## Concept Review
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### Searching After a Hash Collision
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A hash table has 5 buckets and uses the hash function $h(x)=x \bmod 5$. When a collision occurs, elements are placed one after another in a list within that bucket.
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Insert `[1, 6, 11, 7]` in order:
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<!-- numbered-subquestions -->
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1. Write the contents of buckets 0–4.
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2. When searching for 6, which bucket is checked first, and which elements are examined in order?
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3. Based on the bucket contents from Question 1, do later insertions overwrite earlier ones? Explain using this collision-resolution method.
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??? success "Answer"
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1. Because $1\bmod5=6\bmod5=11\bmod5=1$, while $7\bmod5=2$, the buckets are:
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```text
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0: []
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1: [1, 6, 11]
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2: [7]
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3: []
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4: []
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```
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2. A search for 6 first goes to bucket 1, then compares 1 and 6 in order. The target is found on the second comparison.
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3. Equal hash values mean only that the elements go into the same bucket, not that the elements are equal. Separate chaining keeps all colliding elements in the bucket
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and compares them one at a time during a search, so 1, 6, and 11 do not overwrite one another.
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### Where Do Elements Go After a Hash Table Expands?
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A hash table using separate chaining originally has 5 buckets and the hash function $h(x)=x\bmod5$.
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The keys `[1, 6, 11]` are all in bucket 1.
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The table is now expanded to 7 buckets, and the hash function becomes $h(x)=x\bmod7$:
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<!-- numbered-subquestions -->
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1. Calculate the new bucket number for 1, 6, and 11.
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2. Which buckets contain elements after the expansion?
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3. During the expansion, can the list from the old bucket 1 simply be copied into the new bucket 1? Explain using the results from Questions 1 and 2.
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??? success "Answer"
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1. The new bucket numbers are:
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- $1\bmod7=1$;
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- $6\bmod7=6$;
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- $11\bmod7=4$.
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2. Bucket 1 stores 1, bucket 4 stores 11, and bucket 6 stores 6. The three keys are no longer crowded into the same bucket.
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3. The list cannot be copied as is. A bucket number is calculated by taking the key modulo the number of buckets. When the bucket count changes from 5 to 7, a key's bucket number may change,
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so the position of every key must be recalculated. If the old bucket 1 were copied unchanged, later searches using the new formula would go to bucket 6 for 6 and bucket 4 for 11
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and would fail to find them.
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### Can 11 Still Be Found After Deleting 6?
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A hash table has 5 positions with indices `0–4` and uses the hash function $h(x)=x\bmod5$.
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When a collision occurs, it searches to the right from the index produced by the hash function for the first empty position.
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Insert `[1, 6, 11]` in order:
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<!-- numbered-subquestions -->
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1. At which index is each number ultimately stored?
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2. When searching for 11, which indices are examined in order?
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3. Suppose deleting 6 changes its position directly to an "unused empty position," and a search stops whenever it reaches an empty position.
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What happens when searching for 11 afterward? Is this search result correct? If there is a problem, how can it be avoided?
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??? success "Answer"
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1. The number 1 is stored at index 1. The number 6 also maps to index 1, so after the collision it is stored at index 2.
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The number 11 also starts at index 1, skips the occupied indices 1 and 2, and is ultimately stored at index 3.
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2. A search for 11 examines indices `1, 2, 3` in order and finds it at index 3.
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3. If index 2 is changed to mean "never used," a search for 11 checks index 1 and then stops at index 2,
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incorrectly concluding that 11 is absent. Deletion should leave a "deleted" marker.
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A search that reaches this marker continues to the next index (wrapping from index 4 to index 0), while a later insertion may still reuse the position.
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## Programming Exercises
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### Compare the Character Counts of Two Strings
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Given two strings `s` and `t` containing only lowercase English letters,
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you may rearrange the characters in `s` in any order, but you may not add, remove, or replace characters.
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Determine whether the rearranged string can form `t`. Return `true` if it can and `false` otherwise.
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Use a hash table to record how many times each letter occurs. Do not sort the characters in the strings.
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??? tip "Hints"
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1. If the two strings have different lengths, they cannot contain each character the same number of times
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2. Use a hash table to record the count of each letter; increment the corresponding count while scanning s
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3. Decrement the corresponding count while scanning t; the strings contain each character the same number of times only if every count is 0 at the end
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[LeetCode](https://leetcode.com/problems/valid-anagram/){ .rounded-button .exercise-button target="_blank" rel="noopener noreferrer" }
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