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David Luebke 1 6/7/2014 ITCS 6114 Skip Lists Hashing

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David Luebke 2 6/7/2014 Red-Black Trees Red-black trees do what they do very well What do you think is the worst thing about red- black trees? A: coding them up

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David Luebke 3 6/7/2014 Skip Lists A relatively recent data structure A probabilistic alternative to balanced trees A randomized algorithm with benefits of r-b trees O(lg n) expected time for Search, Insert O(1) time for Min, Max, Succ, Pred Much easier to code than r-b trees Fast!

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David Luebke 4 6/7/2014 Linked Lists Think about a linked list as a structure for dynamic sets. What is the running time of: Min() and Max() ? Successor() ? Delete() ? How can we make this O(1)? Predecessor() ? Search() ? Insert() ? Goal: make these O(lg n) time in a linked-list setting So these all take O(1) time in a linked list. Can you think of a way to do these in O(1) time in a red-black tree?

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David Luebke 5 6/7/2014 Skip Lists The basic idea: Keep a doubly-linked list of elements Min, max, successor, predecessor: O(1) time Delete is O(1) time, Insert is O(1)+Search time During insert, add each level-i element to level i+1 with probability p (e.g., p = 1/2 or p = 1/4) level 1 391218293537 level 2 level 3

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David Luebke 6 6/7/2014 Skip List Search To search for an element with a given key: Find location in top list Top list has O(1) elements with high probability Location in this list defines a range of items in next list Drop down a level and recurse O(1) time per level on average O(lg n) levels with high probability Total time: O(lg n)

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David Luebke 7 6/7/2014 Skip List Insert Skip list insert: analysis Do a search for that key Insert element in bottom-level list With probability p, recurse to insert in next level Expected number of lists = 1+ p + p 2 + … = ??? = 1/(1-p) = O(1) if p is constant Total time = Search + O(1) = O(lg n) expected Skip list delete: O(1)

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David Luebke 8 6/7/2014 Skip Lists O(1) expected time for most operations O(lg n) expected time for insert O(n 2 ) time worst case (Why?) But random, so no particular order of insertion evokes worst-case behavior O(n) expected storage requirements (Why?) Easy to code

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David Luebke 9 6/7/2014 Review: Hashing Tables Motivation: symbol tables A compiler uses a symbol table to relate symbols to associated data Symbols: variable names, procedure names, etc. Associated data: memory location, call graph, etc. For a symbol table (also called a dictionary), we care about search, insertion, and deletion We typically dont care about sorted order

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David Luebke 10 6/7/2014 Review: Hash Tables More formally: Given a table T and a record x, with key (= symbol) and satellite data, we need to support: Insert (T, x) Delete (T, x) Search(T, x) We want these to be fast, but dont care about sorting the records The structure we will use is a hash table Supports all the above in O(1) expected time!

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David Luebke 11 6/7/2014 Hashing: Keys In the following discussions we will consider all keys to be (possibly large) natural numbers How can we convert floats to natural numbers for hashing purposes? How can we convert ASCII strings to natural numbers for hashing purposes?

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David Luebke 12 6/7/2014 Review: Direct Addressing Suppose: The range of keys is 0..m-1 Keys are distinct The idea: Set up an array T[0..m-1] in which T[i] = xif x T and key[x] = i T[i] = NULLotherwise This is called a direct-address table Operations take O(1) time! So whats the problem?

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David Luebke 13 6/7/2014 The Problem With Direct Addressing Direct addressing works well when the range m of keys is relatively small But what if the keys are 32-bit integers? Problem 1: direct-address table will have 2 32 entries, more than 4 billion Problem 2: even if memory is not an issue, the time to initialize the elements to NULL may be Solution: map keys to smaller range 0..m-1 This mapping is called a hash function

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David Luebke 14 6/7/2014 Hash Functions Next problem: collision T 0 m - 1 h(k 1 ) h(k 4 ) h(k 2 ) = h(k 5 ) h(k 3 ) k4k4 k2k2 k3k3 k1k1 k5k5 U (universe of keys) K (actual keys)

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David Luebke 15 6/7/2014 Resolving Collisions How can we solve the problem of collisions? Solution 1: chaining Solution 2: open addressing

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David Luebke 16 6/7/2014 Open Addressing Basic idea (details in Section 12.4): To insert: if slot is full, try another slot, …, until an open slot is found (probing) To search, follow same sequence of probes as would be used when inserting the element If reach element with correct key, return it If reach a NULL pointer, element is not in table Good for fixed sets (adding but no deletion) Example: spell checking Table neednt be much bigger than n

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David Luebke 17 6/7/2014 Chaining Chaining puts elements that hash to the same slot in a linked list: T k4k4 k2k2 k3k3 k1k1 k5k5 U (universe of keys) K (actual keys) k6k6 k8k8 k7k7 k1k1 k4k4 k5k5 k2k2 k3k3 k8k8 k6k6 k7k7

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David Luebke 18 6/7/2014 Chaining How do we insert an element? T k4k4 k2k2 k3k3 k1k1 k5k5 U (universe of keys) K (actual keys) k6k6 k8k8 k7k7 k1k1 k4k4 k5k5 k2k2 k3k3 k8k8 k6k6 k7k7

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David Luebke 19 6/7/2014 Chaining T k4k4 k2k2 k3k3 k1k1 k5k5 U (universe of keys) K (actual keys) k6k6 k8k8 k7k7 k1k1 k4k4 k5k5 k2k2 k3k3 k8k8 k6k6 k7k7 How do we delete an element? Do we need a doubly-linked list for efficient delete?

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David Luebke 20 6/7/2014 Chaining How do we search for a element with a given key? T k4k4 k2k2 k3k3 k1k1 k5k5 U (universe of keys) K (actual keys) k6k6 k8k8 k7k7 k1k1 k4k4 k5k5 k2k2 k3k3 k8k8 k6k6 k7k7

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David Luebke 21 6/7/2014 Analysis of Chaining Assume simple uniform hashing: each key in table is equally likely to be hashed to any slot Given n keys and m slots in the table: the load factor = n/m = average # keys per slot What will be the average cost of an unsuccessful search for a key?

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David Luebke 22 6/7/2014 Analysis of Chaining Assume simple uniform hashing: each key in table is equally likely to be hashed to any slot Given n keys and m slots in the table, the load factor = n/m = average # keys per slot What will be the average cost of an unsuccessful search for a key? A: O(1+ )

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David Luebke 23 6/7/2014 Analysis of Chaining Assume simple uniform hashing: each key in table is equally likely to be hashed to any slot Given n keys and m slots in the table, the load factor = n/m = average # keys per slot What will be the average cost of an unsuccessful search for a key? A: O(1+ ) What will be the average cost of a successful search?

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David Luebke 24 6/7/2014 Analysis of Chaining Assume simple uniform hashing: each key in table is equally likely to be hashed to any slot Given n keys and m slots in the table, the load factor = n/m = average # keys per slot What will be the average cost of an unsuccessful search for a key? A: O(1+ ) What will be the average cost of a successful search? A: O(1 + /2) = O(1 + )

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David Luebke 25 6/7/2014 Analysis of Chaining Continued So the cost of searching = O(1 + ) If the number of keys n is proportional to the number of slots in the table, what is ? A: = O(1) In other words, we can make the expected cost of searching constant if we make constant

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David Luebke 26 6/7/2014 Choosing A Hash Function Clearly choosing the hash function well is crucial What will a worst-case hash function do? What will be the time to search in this case? What are desirable features of the hash function? Should distribute keys uniformly into slots Should not depend on patterns in the data

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David Luebke 27 6/7/2014 Hash Functions: The Division Method h(k) = k mod m In words: hash k into a table with m slots using the slot given by the remainder of k divided by m What happens to elements with adjacent values of k? What happens if m is a power of 2 (say 2 P )? What if m is a power of 10? Upshot: pick table size m = prime number not too close to a power of 2 (or 10)

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David Luebke 28 6/7/2014 Hash Functions: The Multiplication Method For a constant A, 0 < A < 1: h(k) = m (kA - kA ) What does this term represent?

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David Luebke 29 6/7/2014 Hash Functions: The Multiplication Method For a constant A, 0 < A < 1: h(k) = m (kA - kA ) Choose m = 2 P Choose A not too close to 0 or 1 Knuth: Good choice for A = ( 5 - 1)/2 Fractional part of kA

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David Luebke 30 6/7/2014 Hash Functions: Worst Case Scenario Scenario: You are given an assignment to implement hashing You will self-grade in pairs, testing and grading your partners implementation In a blatant violation of the honor code, your partner: Analyzes your hash function Picks a sequence of worst-case keys, causing your implementation to take O(n) time to search Whats an honest CS student to do?

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David Luebke 31 6/7/2014 Hash Functions: Universal Hashing As before, when attempting to foil an malicious adversary: randomize the algorithm Universal hashing: pick a hash function randomly in a way that is independent of the keys that are actually going to be stored Guarantees good performance on average, no matter what keys adversary chooses

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