Intro to Computer Algorithms Lecture 16

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Intro to Computer Algorithms Lecture 16

• Phillip G. Bradford
• Computer Science
• University of Alabama

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Announcements

• Advisory Boards Industrial Talk Series
• http//www.cs.ua.edu/9IndustrialSeries.shtm
• Next Research Colloquia
• Prof. Nael Abu-Ghazaleh
• 10-Nov _at_ 1100am
• Active Routing in Mobile Ad Hoc Networks

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CS Story Time

• Prof. Jones research group
• See http//cs.ua.edu/StoryHourSlide.pdf

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Next Midterm

• Tuesday before Thanksgiving !

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Outline

• The Dictionary Problem and Hashing
• Open Hashing with Chaining
• Hashing with Open Addressing
• Double Hashing
• Introduction to Dynamic Programming

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The Dictionary Problem

• Motivation
• Fundamental Operations
• Insert
• Search
• Delete
• Desire Quick Lookup
• Large Records We Focus on the Key

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Hash Tables

• Two desired properties
• Uniform distribution in the table
• Let K be the key to hash and h is the hash
  function
• Table of size m and for any i in 1,,m, then
• Pi h(K) 1/m
• The hash function h is easy to compute

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Hash Tables

• What are some candidates for h?
• h(x) (ax b) mod m
• For appropriate (random?) a and b and mp a prime
• Let Kc0c1cn, characters
• h(x)
• R?0 // and appropriate C
• For i?1 to n do
• R?(RC ord(ci)) mod m
• Endfor
• Return R.

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Hash Tables

• These functions all resemble linear-congruential
  pseudo-random number generators!
• Why is this not surprising?
• Consider two different keys Ki and Kj
• A hash collision is when h(Ki) h(Kj)
• Two approaches to resolve this situation

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Hash Collision Resolution Methods

• Separate Chaining
• Also called Open Hashing
• Given n keys, how long can chains be?
• Worst case?
• Expected length?
• What is the expectation computed over?
• Load Factor a n/m
• In an average chain, when might we expect to find
  an element?
• Successful Searches about 1 a/2 comparisons
• Unsuccessful searches about a comparisons

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Uniform Inputs Balanced Chains

• Chains

NULL
NULL
NULL
NULL
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Hash Collision Resolution Methods

• Linear Probing
• Inserting a key K
• Start at h(K) and if it is full, then try h(K)1,
  h(K)2, etc., until one of these is empty
• How do we know that the hash table is full?
• What can we track?
• U is about (11/(1-a)2)/2
• S is about (11/(1-a))/2

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Hash Collision Resolution Methods

• Double Hashing
• Two pseudo-random functions
• h1(x) and h2(x)
• Use h1(x) to find the starting place
• If done, then stop
• Use h2(x) to determine how much to hop from
  there
• L? h1(x) try position HL first
• Q ? h2(x)
• Otherwise, try positions HL iQ mod m
• For i?1 to floor( m/n )

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Hash Collision Resolution Methods

• U is about 11/(1-a)
• S is about 1/(1-a)

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A Basic Fact

• CLRS2001 IA 1 if A happens
• And IA 0 if A does not happen
• Coin Example XH IX H.
• XH 1 if coin is heads
• XH 0 if coin is tails
• PXH ½
• But also, EXH ½

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A Basic Fact

• Lemma See CLRS2001 Take A in sample space S, if
  XAIA,
• then EXA PA.
• Proof
• EXA 1PA 0PNot(A)
• PA.

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A Look Under the Hood

• Hashing by chaining has average case O(1a)
  probes for a successful search.
• Proof Sketch (full details at the board)
• Assume Pih(K)1/m
• Therefore by the last Lemma Eih(K)1/m

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A Look Under the Hood

• The number of probes in a successful search
• E1/n S(1SXi,j)
• Outer sum i?1 to n
• Inner sum j?i1 to n
• Why?
• Rest at the board
• Double Hashing on Thursday

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Introduction to Dynamic Programming

• Richard Bellman in the 1950s
• Optimizing multistage decision processes
• Problems generally have overlapping subproblems

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Dynamic Programming Example

• Pascals Triangle
• (ab)n
• Classic Relation
• C(n,k) C(n-1,k-1) C(n-1,k)
• Base conditions C(n,0)C(n,n)1.