Lecture 2, Jan 12

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Lecture 2, Jan 12
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CSC373 Algorithm Design and Analysis

• Office hours Wed 5-6 pm, starting next week.
• Tutorials start today 8-9pm.
• Assignment 1.

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Review

• Last time have seen the interval scheduling
  problem.

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Interval Scheduling

• Ingredients
• Instances Activities with starting and finishing
  times ltlts1,f1gt,lts2,f2gt, ,ltsn,fngtgt.
• Solutions A set of events that do not overlap.
• Cost of Solution The number of activities
  scheduled.
• Goal Given a set of activities, schedule as many
  as possible.

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The greedy algorithm
Greedy Criteria
Earliest Finishing Time
Schedule the event who will free up your room
for someone else as soon as possible.
Motivation
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Outline of the algorithm

• Order the activities in non-decreasing order of
  finishing times. Denote it by A1, A2, , An.
• For i1,,n
• If Ai does not conflict with previous choices,
  include it in the schedule.

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Outline of the proof generic greedy algorithm

• Since there is not backtracking in the greedy
  algorithms, we must show that the algorithm never
  makes irreversible mistakes.
• In other words, the partial solution the
  algorithm currently holds can be extended to an
  optimal solution.
• At the end, when the algorithm holds a solution,
  it must be optimal (a solution is a partial
  solution).

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Generic greedy algorithms contd

• We say that a partial solution is promising, if
  it can be extended to an optimal solution.
• Prove by induction the loop invariant
• any partial solution the algorithm holds is
  promising
• In the interval scheduling case, we proved that
  the interval selections Si for the first i
  intervals is promising.

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Generic greedy algorithms contd

• The induction step is usually done using a
  replacement argument.
• Massage the optimal solution extending the i-th
  step to extend the i1-st step.
• For interval scheduling, if the solution Si was
  not extending Si1, it was modified by replacing
  some interval in Si by an interval the algorithm
  has chosen to obtain Si1.

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Generic greedy algorithms contd

• Finally, use the fact that the last partial
  solution is promising to show that the algorithm
  works.
• For interval scheduling, the solution Sn (which
  includes all the choices that algorithm made
  about all the intervals) is promising, and hence
  is optimal.

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Minimum Spanning Trees

• Example Problem
• You are planning a new terrestrial
  telecommunications network to connect a number of
  remote mountain villages in a developing country.
  
• The cost of building a link between pairs of
  neighboring villages (u,v) has been estimated
  w(u,v).
• You seek the minimum cost design that ensures
  each village is connected to the network.
• The solution is called a minimum spanning tree.

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Properties of a Minimum Spanning Tree

• V-1 edges
• Acyclic
• Not necessarily unique

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A greedy strategy for MST Kruskals algorithm

• The idea
• expand the network at minimal cost
• Throw in the lightest edge that does not close a
  cycle

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Example

• Kruskals Algorithm

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Kruskals Algorithm for computing MST
O(ElogE)
Running Time
O(ElogV)
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Kruskals algorithm correctness proof plan

• The algorithm obviously terminates
• show that at each step the partial solution the
  algorithm holds is promising
• can be extended to an MST
• when the algorithm terminates, show that the
  solution it holds is a tree
• it is promising, hence it must be an MST.

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Prims Algorithm

• A different greedy strategy.
• Build the net by choosing the cheapest link to
  connect next.

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Prims Algorithm contd

• Build one tree
• Start from arbitrary root r
• At each step, add lightest edge crossing cut (VA,
  V- VA), where VA vertices processed so far.

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Finding light edges quickly

• Use priority queue
• Each object in the queue is a vertex v that is
  still waiting to be connected (in V-VA)
• Key of v is minimum weight of any edge (u,v), u?
  VA.
• Each vertex knows its parent in tree by ?v.

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Prims Algorithm
O(Elog(V))
Running Time
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Prims algorithm correctness proof plan

• The algorithm obviously terminates
• show that at each step the partial solution the
  algorithm holds is promising
• can be extended to an MST
• when the algorithm terminates, show that the
  solution it holds is a tree
• it is promising, hence it must be an MST.

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Three Knapsack Examples

• 0-1 knapsack with variable value/weight
• 0-1 knapsack with fixed value/weight
• Fractional knapsack with variable value/weight