ICS 353: Design and Analysis of Algorithms

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ICS 353 Design and Analysis of Algorithms
King Fahd University of Petroleum
Minerals Information Computer Science Department

• Greedy Algorithms

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Reading Assignment

• M. Alsuwaiyel, Introduction to Algorithms Design
  Techniques and Analysis, World Scientific
  Publishing Co., Inc. 1999.
• Chapter 8 Section 1.
• Chapter 8 Sections 8.2 8.4 (Except 8.2.1 and
  8.4.1)
• T. Cormen, C. Leiserson, R. Rivest C. Stein
  Introduction to Algorithms, 2nd Edition, The MIT
  Press, 2001.

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Greedy Algorithms

• Like dynamic programming algorithms, greedy
  algorithms are usually designed to solve
  optimization problems
• Unlike dynamic programming algorithms,
• greedy algorithms are iterative in nature.
• An optimal solution is reached from local optimal
  solutions.
• This approach does not work all the time.
• A proof that the algorithm does what it claims is
  needed, and usually not easy to get.

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Fractional Knapsack Problem

• Given n items of sizes s1, s2, , sn and values
  v1, v2, , vn and size C, the problem is to find
  x1, x2, , xn ?? that maximize
• subject to

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Solution to Fractional Knapsack Problem

• Consider yi vi / si
• What is yi?
• What is the solution?

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Activity Selection Problem

• Problem Formulation
• Given a set of n activities, S a1, a2, ...,
  an that require exclusive use of a common
  resource, find the largest possible set of
  nonoverlapping activities (also called mutually
  compatible).
• For example, scheduling the use of a classroom.
• Assume that ai needs the resource during period
  si, fi), which is a half-open interval, where
• si start time of the activity, and
• fi finish time of the activity.
• Note Could have many other objectives
• Schedule room for longest time.
• Maximize income rental fees.

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Activity Selection Problem Example

• Assume the following set S of activities that are
  sorted by their finish time, find a maximum-size
  mutually compatible set.

i 1 2 3 4 5 6 7 8 9
si 1 2 4 1 5 8 9 11 13
fi 3 5 7 8 9 10 11 14 16
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i 1 2 3 4 5 6 7 8 9
si 1 2 4 1 5 8 9 11 13
fi 3 5 7 8 9 10 11 14 16
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Solving the Activity Selection Problem

• Define Si,j ak ?? S fi ? sk lt fk ? sj
• activities that start after ai finishes and
  finish before aj starts
• Activities in Si,j are compatible with
• Add the following fictitious activities
• a0 ?? , 0) and an1 ? , ?1)
• Hence, S S0,n1 and
• the range of Si,j is 0 ? i,j ? n1

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Solving the Activity Selection Problem

• Assume that activities are sorted by
  monotonically increasing finish time
• i.e., f0 ? f1 ? f2 ? ... ? fn lt fn1
• Then, Si,j ?? for i ? j.
• Proof
• Therefore, we only need to worry about Si,j where
  0 ? i lt j ? n1

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Solving the Activity Selection Problem

• Suppose that a solution to Si,j includes ak . We
  have 2 sub-problems
• Si,k (start after ai finishes, finish before ak
  starts)
• Sk,j (start after ak finishes, finish before aj
  starts)
• The Solution to Si,j is
• (solution to Si,k ) ?? ak ? (solution to
  Sk,j )
• Since ak is in neither sub-problem, and the
  subproblems are disjoint,
• solution to S solution to
  Si,k1solution to Sk,j

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Recursive Solution to Activity Selection Problem

• Let Ai,j optimal solution to Si,j .
• So Ai,j Ai,k ? ak ? Ak,j, assuming
• Si,j is nonempty, and
• we know ak.
• Hence,

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Finding the Greedy Algorithm

• Theorem Let Si,j ? ?, and let am be the activity
  in Si,j with the earliest finish time fm min
  fk ak ? Si,j . Then
• am is used in some maximum-size subset of
  mutually compatible activities of Si,j
• Sim ?, so that choosing am leaves Sm,j as the
  only nonempty subproblem.

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Recursive Greedy Algorithm
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Iterative Greedy Algorithm
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Greedy Strategy

1. Determine the optimal substructure.
2. Develop a recursive solution.
3. Prove that at any stage of recursion, one of the
   optimal choices is the greedy choice. Therefore,
   it's always safe to make the greedy choice.
4. Show that all but one of the subproblems
   resulting from the greedy choice are empty.
5. Develop a recursive greedy algorithm.
6. Convert it to an iterative algorithm.

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Shortest Paths Problems

• Input A graph with non-negative weights or costs
  associated with each edge.
• Output The list of edges forming the shortest
  path.
• Sample problems
• Find shortest path between two named vertices
• Find shortest path from S to all other vertices
• Find shortest path between all pairs of vertices
• Will actually calculate only distances, not paths.

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Shortest Paths Definitions

• ?(A, B) is the shortest distance from vertex A to
  B.
• length(A, B) is the weight of the edge connecting
  A to B.
• If there is no such edge, then length(A, B) ?.

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A
C
1
10
5
B
D
7
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Single-Source Shortest Paths

• Problem Given G(V,E), start vertex s, find the
  shortest path from s to all other vertices.
• Assume V1, 2, , n and s1
• Solution A greedy algorithm called Dijkstras
  Algorithm

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Dijkstras Algorithm Outline

• Partition V into two sets
• X1 and Y 2, 3, , n
• Initialize ?i for 1 ? i ? n as follows
• Select y?Y such that ?y is minimum
• ?y is the length of the shortest path from 1 to
  y that uses only vertices in set X.
• Remove y from Y, add it to X, and update ?w for
  each w?Y where (y,w) ? E if the path through y is
  shorter.

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Example
5
6
1
2
11
2
3
15
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Dijkstras Algorithm

A B C D E
Initial 0 ? ? ? ?
Process A
Process
Process
Process
Process
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Dijkstras Algorithm

• Input A weighted directed graph G (V,E), where
  V 1,2,,n
• Output The distance from vertex 1 to every other
  vertex in G.
• X 1 Y 2,3,,n ?10
• for y 2 to n do
• if y is adjacent to 1 then ?ylength1,y
• else ?y ? end if
• end for
• for j 1 to n 1 do
• Let y ? Y be such that ?y is minimum
• X X ?? y // add vertex y to X
• Y Y ?- y // delete vertex y from Y
• for each edge (y,w) do
• if w ? Y and ?y lengthy,w lt ?w then
• ?w ?y lengthy,w
• end for
• end for

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Correctness of Dijkstras Algorithm

• Lemma In Dijkstras algorithm, when a vertex y
  is chosen in Step 7, if its label ?y is finite
  then ?y ?? y.
• Proof

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Time Complexity

• Mainly depends on how we implement step 7, i.e.,
  finding y s.t. ?y is minimum.
• Approach 1 Scan through the vector representing
  current distances of vertices in Y
• Approach 2 Use a min-heap to maintain vertices
  in the set Y

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Minimum Cost Spanning Trees

• Minimum Cost Spanning Tree (MST) Problem
• Input An undirected weighted connected graph G.
• Output The subgraph of G that
• 1) has minimum total cost as measured by
  summing the weights of all the edges in the
  subset, and
• 2) keeps the vertices connected.
• What does such subgraph look like?

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MST Example
5
2
1
4
11
2
3
2
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Kruskals Algorithm

• Initially, each vertex is in its own MST.
• Merge two MSTs that have the shortest edge
  between them.
• Use a priority queue to order the unprocessed
  edges. Grab next one at each step.
• How to tell if an edge connects two vertices that
  are already in the same MST?
• Use the UNION/FIND algorithm with parent-pointer
  representation.

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Example
5
2
1
3
11
2
3
2
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Kruskals MST Algorithm

• Sort the edges of E(G) by weight in
  non-decreasing order
• For each vertex v ? V(G) do
• New_Tree(v) // creating tree with one root
  node v
• T? // MST initialized to empty
• While T lt n - 1 do
• Let (u,v) be the next edge in E(G)
• if FIND(u) ? FIND(v) then
• T T ? (u,v)
• UNION(u,v)
• Return T

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Asymptotic Analysis of Kruskals Algorithm
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Correctness of Kruskals Algorithm

• Lemma Algorithm Kruskal correctly finds a
  minimum cost spanning tree in a weighted
  undirected graph.
• Proof
• Theorem Algorithm Kruskals finds a minimum cost
  spanning tree in a weighted undirected graph in
  O(m log m) time, where m is the number of edges
  in G.

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Money Change Problem

• Given a currency system that has n coins with
  values v1, v2 , ..., vn, where v1 1, the
  objective is to pay change of value y in such a
  way that the total number of coins is minimized.
  More formally, we want to minimize the quantity
• subject to the constraint
• Here, x1, x2 , ..., xn, are nonnegative integers
  (so xi may be zero).

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Money Change Problem

• What is a greedy algorithm to solve this problem?
• Is the greedy algorithm optimal?