Lecture 2: The Greedy Method

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Lecture 2 The Greedy Method

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Content

• What is it?
• Activity Selection Problem
• Fractional Knapsack Problem
• Minimum Spanning Tree
• Kruskals Algorithm
• Prims Algorithm
• Shortest Path Problem
• Dijkstras Algorithm
• Huffman Codes

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Lecture 2 The Greedy Method

• What is it?

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The Greedy Method

• A greedy algorithm always makes the choice that
  looks best at the moment
• For some problems, it always give a globally
  optimal solution.
• For others, it may only give a locally optimal
  one.

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Main Components

• Configurations
• different choices, collections, or values to find
• Objective function
• a score assigned to configurations, which we want
  to either maximize or minimize

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Example Making Change
Is the solution always optimal?

• Problem
• A dollar amount to reach and a collection of coin
  amounts to use to get there.
• Configuration
• A dollar amount yet to return to a customer plus
  the coins already returned
• Objective function
• Minimize number of coins returned.
• Greedy solution
• Always return the largest coin you can

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Example Largest k-out-of-n Sum

• Problem
• Pick k numbers out of n numbers such that the sum
  of these k numbers is the largest.
• Exhaustive solution
• There are choices.
• Choose the one with subset sum being the largest
• Greedy Solution
• FOR i 1 to k
• pick out the largest number and
• delete this number from the input.
• ENDFOR

Is the greedy solution always optimal?
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ExampleShortest Paths on a Special Graph

• Problem
• Find a shortest path from v0 to v3
• Greedy Solution

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ExampleShortest Paths on a Special Graph
Is the solution optimal?

• Problem
• Find a shortest path from v0 to v3
• Greedy Solution

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ExampleShortest Paths on a Multi-stage Graph
Is the greedy solution optimal?

• Problem
• Find a shortest path from v0 to v3

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ExampleShortest Paths on a Multi-stage Graph
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Is the greedy solution optimal?

• Problem
• Find a shortest path from v0 to v3

The optimal path
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ExampleShortest Paths on a Multi-stage Graph
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Is the greedy solution optimal?

• Problem
• Find a shortest path from v0 to v3

What algorithm can be used to find the optimum?
The optimal path
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Advantage and Disadvantageof the Greedy Method

• Advantage
• Simple
• Work fast when they work
• Disadvantage
• Not always work ? Short term solutions can be
  disastrous in the long term
• Hard to prove correct

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Lecture 2 The Greedy Method

• Activity Selection Problem

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Activity Selection Problem(Conference Scheduling
Problem)

• Input A set of activities S a1,, an
• Each activity has a start time and a finish time
• ai si, fi)
• Two activities are compatible if and only if
  their interval does not overlap
• Output a maximum-size subset of mutually
  compatible activities

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ExampleActivity Selection Problem
Assume that fis are sorted.
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ExampleActivity Selection Problem
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ExampleActivity Selection Problem
Is the solution optimal?
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ExampleActivity Selection Problem
Is the solution optimal?
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Activity Selection Algorithm
Greedy-Activity-Selector (s, f) // Assume that
f1 ? f2  ?   ...  ? fn  n ? length s A ?
1 j ? 1 for i ? 2 to n    if si ? fj
then A ? A ? i j ? i return A
Is the algorithm optimal?
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Proof of Optimality

• Suppose A ? S is an optimal solution and the
  first activity is k ? 1.
• If k ? 1, one can easily show that B A k ?
  1 is also optimal. (why?)
• This reveals that greedy-choice can be applied to
  the first choice.
• Now, the problem is reduced to activity selection
  on S 2, , n, which are all compatible with
  1.
• By the same argument, we can show that, to retain
  optimality, greedy-choice can also be applied for
  next choices.

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Lecture 2 The Greedy Method

• Fractional Knapsack Problem

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The Fractional Knapsack Problem

• Given A set S of n items, with each item i
  having
• bi - a positive benefit
• wi - a positive weight
• Goal Choose items, allowing fractional amounts,
  to maximize total benefit but with weight at most
  W.

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The Fractional Knapsack Problem
knapsack

• Solution
• 1 ml of 5
• 2 ml of 3
• 6 ml of 4
• 1 ml of 2

Items
wi
4 ml
8 ml
2 ml
6 ml
1 ml
bi
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The Fractional Knapsack Algorithm

• Greedy choice Keep taking item with highest value

Algorithm fractionalKnapsack(S, W) Input set S
of items w/ benefit bi and weight wi max.
weight W Output amount xi of each item i to
maximize benefit w/ weight at most W for each
item i in S xi ? 0 vi ? bi / wi value w ?
0 total weight while w lt W remove item i
with highest vi xi ? minwi , W ? w w ? w
minwi , W ? w
Does the algorithm always gives an optimum?
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Proof of Optimality

• Suppose there is a better solution
• Then, there is an item i with higher value than a
  chosen item j, but xi lt wi, xj gt 0 and vi gt vj
• Substituting some i with j, well get a better
  solution
• How much of i minwi ? xi, xj
• Thus, there is no better solution than the greedy
  one

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Recall 0-1 Knapsack Problem
Which boxes should be chosen to maximize the
amount of money while still keeping the overall
weight under 15 kg ?
Is the fractional knapsack algorithm applicable?
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Exercise

• Construct an example show that the fractional
  knapsack algorithm doesnt give the optimal
  solution when applying it to the 0-1 knapsack
  problem.

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Lecture 2 The Greedy Method

• Minimum
• Spanning Tree

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What is a Spanning Tree?

• A tree is a connected undirected graph that
  contains no cycles
• A spanning tree of a graph G is a subgraph of G
  that is a tree and contains all the vertices of G

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Properties of a Spanning Tree

• The spanning tree of a n-vertex undirected graph
  has exactly n 1 edges
• It connects all the vertices in the graph
• A spanning tree has no cycles

Undirected Graph
Some Spanning Trees
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What is a Minimum Spanning Tree?

• A spanning tree of a graph G is a subgraph of G
  that is a tree and contains all the vertices of G
• A minimum spanning tree is the one among all the
  spanning trees with the lowest cost

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Applications of MSTs

• Computer Networks
• To find how to connect a set of computers using
  the minimum amount of wire
• Shipping/Airplane Lines
• To find the fastest way between locations

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Two Greedy Algorithms for MST

• Kruskals Algorithm
• merges forests into tree by adding small-cost
  edges repeatedly
• Prims Algorithm
• attaches vertices to a partially built tree by
  adding small-cost edges repeatedly

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Kruskals Algorithm
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Kruskals Algorithm
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Kruskals Algorithm
G (V, E) Graph w E?R Weight T ? Tree

• MST-Kruksal(G)
• T ? Ø
• for each vertex v ? VG
• Make-Set(v) // Make separate sets for vertices
• sort the edges by increasing weight w
• for each edge (u, v) ? E, in sorted order
• if Find-Set(u) ? Find-Set(v) // If no cycles
  are formed
• T ? T ? (u, v) // Add edge to Tree
• Union(u, v) // Combine Sets
• return T

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Time Complexity
O(ElogE)
G (V, E) Graph w E?R Weight T ? Tree

• MST-Kruksal(G , w)
• T ? Ø
• for each vertex v ? VG
• Make-Set(v) // Make separate sets for vertices
• sort the edges by increasing weight w
• for each edge (u, v) ? E, in sorted order
• if Find-Set(u) ? Find-Set(v) // If no cycles
  are formed
• T ? T ? (u, v) // Add edge to Tree
• Union(u, v) // Combine Sets
• return T

O(1)
O(V)
O(ElogE)
O(E)
O(V)
O(1)
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Prims Algorithm
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Prims Algorithm
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Prims Algorithm
G (V, E) Graph w E?R Weight r Starting
vertex Q Priority Queue Keyv Key of Vertex
v pv Parent of Vertex v Adjv Adjacency
List of v

• MST-Prim(G, w, r)
• Q ? VG // Initially Q holds all vertices
• for each u ? Q
• Keyu ? 8 // Initialize all Keys to 8
• Keyr ? 0 // r is the first tree node
• pr ? Nil
• while Q ? Ø
• u ? Extract_min(Q) // Get the min key node
• for each v ? Adju
• if v ? Q and w(u, v) lt Keyv // If the
  weight is less than the Key
• pv ? u
• Keyv ? w(u, v)

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Time Complexity
O(ElogV)
G (V, E) Graph w E?R Weight r Starting
vertex Q Priority Queue Keyv Key of Vertex
v pv Parent of Vertex v Adjv Adjacency
List of v

• MST-Prim(G, r)
• Q ? VG // Initially Q holds all vertices
• for each u ? Q
• Keyu ? 8 // Initialize all Keys to 8
• Keyr ? 0 // r is the first tree node
• pr ? Nil
• while Q ? Ø
• u ? Extract_min(Q) // Get the min key node
• for each v ? Adju
• if v ? Q and w(u, v) lt Keyv // If the
  weight is less than the Key
• pv ? u
• Keyv ? w(u, v)

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Optimality
Are the algorithms optimal?
Yes

• Kruskals Algorithm
• merges forests into tree by adding small-cost
  edges repeatedly
• Prims Algorithm
• attaches vertices to a partially built tree by
  adding small-cost edges repeatedly

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Lecture 2 The Greedy Method

• Shortest Path Problem

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Shortest Path Problem (SPP)

• Single-Source SPP
• Given a graph G (V, E), and weight w E?R,
  find the shortest path from a source node s ? V
  to any other node, say, v ? V.
• All-Pairs SPP
• Given a graph G (V, E), and weight w E?R,
  find the shortest path between each pair of nodes
  in G.

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Dijkstra's Algorithm

• Dijkstra's algorithm, named after its discoverer,
  Dutch computer scientist Edsger Dijkstra, is an
  algorithm that solves the single-source shortest
  path problem for a directed graph with
  nonnegative edge weights.

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Dijkstra's Algorithm

• Start from the source vertex, s
• Take the adjacent nodes and update the current
  shortest distance
• Select the vertex with the shortest distance,
  from the remaining vertices
• Update the current shortest distance of the
  Adjacent Vertices where necessary,
• i.e. when the new distance is less than the
  existing value
• Stop when all the vertices are checked

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Dijkstra's Algorithm
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Dijkstra's Algorithm
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Dijkstra's Algorithm
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Dijkstra's Algorithm
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Dijkstra's Algorithm
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Dijkstra's Algorithm
G (V, E) Graph w E?R Weight s
Source dv Current shortest distance from s to
v S Set of nodes whose shortest distance is
known Q Set of nodes whose shortest distance is
unknown

• Dijkstra(G, w ,s)
• for each vertex v ? VG
• dv ? ? // Initialize all distances to ?
• pv ? Nil
• ds ? 0 // Set distance of source to 0
• S ? ?
• Q ? VG
• while Q ? ?
• u ? Extract_Min(Q) // Get the min in Q
• S ? S ? u // Add it to the already known
  list
• for each vertex v ? Adju
• if dv gt du w(u, v) // If the new
  distance is shorter
• dv ? du w(u, v)
• pv ? u

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Lecture 2 The Greedy Method

• Huffman Codes

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Huffman Codes

• Huffman code is a technique for compressing 
  data.
• Variable-Length code
• Huffman's greedy algorithm look at the occurrence
  of each character and it as a binary string in an
  optimal way.

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Example
Suppose we have a data consists of 100,000
characters with following frequencies.
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Fixed vs. Variable Length Codes
Suppose we have a data consists of 100,000
characters with following frequencies.
Total Bits
Fixed Length Code
3?45,000 3?13,000 3?12,000 3?16,000
3?9,000 3?5,000 300,000
Variable Length Code
1?45,000 3?13,000 3?12,000 3?16,000
4?9,000 4?5,000 224,000
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Prefix Codes
In which no codeword is a prefix of other
codeword.
Encode
aceabfd
0100110101011100111
Decode
0100110101011100111
a
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Huffman-Code Algorithm
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Huffman-Code Algorithm
a45
c12
b13
d16
f5
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
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Huffman-Code Algorithm
a45
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Huffman-Code Algorithm
Huffman tree built
a45
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Huffman-Code Algorithm
Huffman (C) n ? C Q ? C for i ? 1 to n ? 1
   z ? Allocate-Node ()       x ? leftz ?
Extract-Min (Q) // least frequent       y ?
rightz ? Extract-Min (Q) // next least      
fz ? fx fy // update frequency      
Insert ( Q, z ) return Extract-Min (Q)
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Optimality
Exercise