UCSD CSE 21, Spring 2014 [Section B00] Mathematics for...

UCSD CSE 21, Spring 2014 [Section B00]

Mathematics for Algorithm and System Analysis

Lecture 17

Class URL: http://vlsicad.ucsd.edu/courses/cse21-s14/

Lecture 17 Notes • Last time

• Recurrences revisited• Types of graphs: Complete, Tree, Bipartite, …

• Today and Next Tuesday• Shortest path tree, minimum spanning tree, search trees• Chromatic number• Paths, Trails, Circuits, Cycles: Eulerian and Hamiltonian graphs• (“complexity zoo” – NP, P, NP-hard, NP-complete, …)• (most of the slides are in this slide deck already – please read ahead!)

• Logistics– Final exam review–Sunday June 8, 6-8pm, PCYNH 106

• Practice final exam questions – early Week 10– Week 9 and Week 10 quizzes off-line

• Today’s quiz: open on WeBWorK from 9:30am – 10pm• Can get up to “5 points out of 4”

– Extra credit for PPT slide(s) (up to 2% of course grade)• See next slide

– “Things to Know” – end of Week 9, end of Week 10• SPIS slides on “Trees, Trees, Trees, …”

– Please treat as “assigned reading”

Extra Credit Opportunity: PPT slide(s) on graph(s)– 1% course grade for each slide, up to a max of 2– See thread on Piazza – Must use the given PPT template, and include:

• Image• Source / attribution (URL)• Brief description (what do vertices / edges represent, is the graph

directed / undirected, etc.)• Brief explanation of why the image and the graph are interesting

– Deadline = Thursday, June 5, 5pm Pacific time• Turn-in: email .pptx/.ppt attachment to natalie.lynn.larson@gmail.com• Filename: LastName_FirstName_LastTwoDigitsofPID[_2].pptx

Special kinds of graphs – complete / clique• Complete (simple) graphs: all possible edges

http://en.wikipedia.org/wiki/File:11-simplex_graph.svghttp://en.wikipedia.org/wiki/File:3-simplex_graph.svghttp://en.wikipedia.org/wiki/File:7-simplex_graph.svg

Special kinds of graphs – bipartite • Bipartite graphs

– Two “kinds” of vertices V1 V2 = V, V1 V2 =

– Each edge is between a vertex of one kind and a vertex of the other kind: any e E has form {u V1 , v V2}

People Projects

Special kinds of graphs – tree • Trees

– Connected– |V| - 1 edges – No cycles

(any two of these properties implies the third !)

Assignment: read the SPIS-2013 “Trees, Trees, Trees” slides that are posted …!

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Trees vs. General Graphs

Graph: G = (V,E)

V = set of vertices

E = set of edges

graph with 7 vertices, 6 edges

This graph is not connected, and it has a cycle.

Connected: Between any pair of vertices, there is a path of edges.Cycle: A path of edges that begins and ends at the same vertex.

Tree = Special Kind of Graph

tree with 7 vertices, 6 edges

A tree is a graph with n vertices that:

(i) is connected

(ii) has no cycles

(iii) has n – 1 edges

Fact: Any two of the above properties implies the third.

E.g., “If a graph with n vertices is connected and has no cycles, then it has n – 1 edges.” Can we prove this???

“You want proof? I’ll give you proof!”

Proof Strategy: ?

Claim: If a graph with n vertices is connected and has no cycles, then it has n – 1 edges.

(I.H.) Assume that the Claim is true for n = 1, …, k – 1. Consider a graph with n = k vertices that is connected and has no cycles.

Strong Induction

k1 k – 1 vertices, connected, no cycles k1 – 1 edges

k2 k – 1 vertices, connected, no cycles k2 – 1 edges

(k1 – 1) + (k2 – 1) + 1 = (k1 + k2) – 1 = k – 1

Trees = Data Structures

Binary Search Tree

Application: Search a set of keys for a given key value

if key = root.key then success

else if key < root.key then search left subtree

else search right subtree

Binary tree:

Has a root vertex

Every vertex has at most 2 children

Search for “2”Search for “17”Search for “8”

How Many Search Steps Are Needed?How many levels can a binary tree with n nodes have?

(convention: root is at level 0)

At most: ?

At least: ?

n – 1 levels

log2n levels

Trees = Data Structures = Algorithms

• A heap is a binary tree of depth d such that– (1) all nodes not at depth d-1 or d are internal nodes each level is filled before the next level is started

– (2) at depth d-1 the internal nodes are to the left of the leaves and have degree 2, except perhaps for the rightmost, which has a left child each level is filled left to right

• A max heap (min heap) is a heap with node labels from an ordered set, such that the label at any internal node is () the label of any of its children All root-leaf paths are monotone

Minheap Example Data Structure

• Parent-child arithmetic = how we’d store in an array … !

– Parent parent of A[i] has index = i div 2– Left, Right (children) children of A[i] have indices 2i, 2i+1

• Parent Child this is a minheap

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Heap Operation: Insert

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Insert(S,x): O(height) O(log n)Keep swapping with parent until heap condition satisfied

Heap Operation: ExtractMin

ExtractMin(S): return head, replace head key with the last key, “float down” O(log n)

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“Float down”: If heap condition violated, swap with smaller child

Insert + ExtractMin = ?

• “Priority queue” abstract data type

– Always have the highest-priority item ready at hand– Useful in algorithm implementation – e.g., Dijkstra’s

algorithm later today

• Unordered list: Insert = O(1) but ExtractMin = O(n) • Ordered list: ExtractMin = O(1) but Insert = O(n)

• Heap: Insert = O(log n), ExtractMin = O(log n)

O(n log n) Heapsort• Build heap: n x O(log n) time

– for i = 1..n do insert (A[1..i],A[i])• Extract elements in sorted order: n x O(log n) time

– for i = n..2 do [extract-min]• Swap (A[1] A[i])• Heapsize = Heapsize-1• Float down A[1]

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Trees = Solutions to Real Problems

(A) Connecting cities with minimum road construction

(when adding new junctions is allowed)

(B) Calculating minimum-cost routes to destinations

The Minimum Spanning Tree Problem

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Given: graph of “cities” and “distances”

Problem: Make all cities reachable from each other with minimum road construction.

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The Shortest Paths Problem

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Problem: Compute the shortest routes from A to every other city.

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The Shortest Paths Problem

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Problem: Compute the shortest routes from B to every other city.

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What do these green numbers represent?

The Shortest Paths Problem

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“Exploration parties set out from B, always moving at constant speed. When a party is the first to reach some city, they call home, then split up and continue exploring. The times of the phone calls are the shortest-path distances from B.”

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(You learn this as Dijkstra’s Algorithm)

Minimum Spanning Tree, Again

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“Start from an arbitrary vertex, and add the shortest incident edge to the growing tree that does not complete a cycle.” (If the MST is unique, then the starting vertex doesn’t matter…)(You learn this as Prim’s Algorithm)

Nearly the Same Algorithm !• Prim: Iteratively add the edge e(u,v) to T,

where u is in T and v is not yet in T, tominimize du,v (du,v = (u,v) distance)

• Dijkstra: Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, tominimize du,v + lu (lu is u’s shortest path cost)

• Both algorithms build trees, in similar greedy ways!– Prim’s Algorithm constructs a Minimum Spanning Tree– Dijkstra’s Algorithm constructs a Shortest Path Tree

If Prim’s Algorithm and Dijkstra’s Algorithm had a baby… Iteratively add the edge e(u,v) to T, where u is in T and v is not yet in T, tominimize du,v + c lu // 0 c 1 (food for thought !)

Minimum Spanning Tree = A Greedy Problem

• Prim’s algorithm. Start with T = a root node s, and greedily grow the tree T. At each step, add to T the cheapest edge e that has exactly one endpoint in T.

• Reverse-Delete algorithm. Start with T = E (all edges). Consider edges in descending order of cost. Delete edge e from T unless doing so disconnects T.

• Kruskal’s algorithm. Start with T = . Consider edges in ascending order of cost. Insert edge e into T unless doing so creates a cycle.

• All three methods are greedy, and produce a minimum spanning tree.

Kruskal’s Algorithm

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From Spanning Trees to Steiner Trees

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• The red dot is called a Steiner point (= an “intermediate junction”)• The maximum cost savings from adding Steiner points

= min ratio of (min Steiner tree cost) / (min spanning tree cost)depends on the metric, or distance function

• Cf. “Manhattan”, “Euclidean”, “Chebyshev”, … Adapted from Prof. G. Robins, UVA

Minimum Steiner Tree is a Hard Problem (Minimum Spanning Tree was Easy)

• You learn about “hard” vs. “easy” in CSE101, CSE105• How would you approach finding a low-cost Steiner tree?

Adapted from Prof. G. Robins, UVA

Iterated 1-Steiner Algorithm (1990)

Given a pointset S, what point p minimizes MST(S {p})

Algorithmic idea: Iterate! (greedily)

In practice: solution cost is within 0.5% of OPT on average

Adapted from Prof. G. Robins, UVA

Walks - DefinitionLet G = (V,E,φ) be a graph.

A walk in G is a sequence:

e1, e2, ..., en−1

of elements of E (edges of G) for which there is a sequence:

a1, a2, ..., an

of elements of V (vertices of G) such that φ(ei) = {ai, ai+1} for

i = 1, 2, ..., n−1.

Walks on Graphs

Which of the following are walks on the graph?A. a, b, c, dB. 1,2,3,4C. d,b,a,c,dD. a,1,bE. None of the above / more than one of the above

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Special Walks• (Neither edges nor vertices need be distinct: walk)

• Edges must be distinct: trail

• Vertices must be distinct: path

– Sequence of vertices is the “vertex sequence of the path”

• Trail whose first vertex is same as last vertex: circuit

• Circuit with no repeated vertices (except start/end): cycle

• If the graph is directed:

– Obtain a directed walk, trail, path, circuit, cycle…

– Replace {} with () in vertex set: φ(ei) = (ai, ai+1) now

– Arrows: domain set on the left, codomain set on right

Example

Which of the following statements are true?

A. There are no cycles in this graph.

B. There are no circuits in this graph.

C. There is a path that visits each vertex exactly once.

D. There is a trail that uses each edge exactly once.

E. None of the above/ more than one of the above.

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More on graphs…• A graph is connected if there is a path between every pair of

vertices. – Edge case: graph with only one vertex, we say it is connected

• One vertex is reachable from another if there is a path between them.

• A connected component of a graph is a subset of vertices such that (i) every pair of vertices in the set is reachable from one another, and (ii) the subset is not part of a larger set with this property.

– Computing distance in a graph: breadth first search of all walks• Small world phenomenon in social networks

– Analyze structure of the graph by looking at number of paths• Hubs and spokes• Link prediction in social networks

Visiting “everywhere” on a graph• An Eulerian trail is a trail that includes every edge.

• An Eulerian circuit is a circuit that includes every edge.

Visiting “everywhere” on a graph• An Eulerian trail is a trail that includes every edge.

• An Eulerian circuit is a circuit that includes every edge.

In the previous example there is no Eulerian trail:

How do we prove this?

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Visiting “everywhere” on a graph• An Eulerian trail is a trail that includes every edge.

• An Eulerian circuit is a circuit that includes every edge.

• Observations:– If a graph has an Eulerian circuit then it has an Eulerian trail.– If a graph is not connected, then it does not have an Eulerian trail.

Characterization of Eulerian graphs• Theorem. For a connected simple graph G, the following are

equivalent: – G contains an Eulerian circuit.– Every vertex of G has even degree.– The edges of G can be partitioned into edge-disjoint cycles.

• Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two.

• UCSD connection: “An Eulerian path approach to DNA fragment assembly” Pavel A. Pevzner, Haixu Tang, and Michael S. Waterman (2001).

Characterization of Eulerian graphs• Theorem. For a connected simple graph G, the following are

equivalent: – G contains an Eulerian circuit.– Every vertex of G has even degree.– The edges of G can be partitioned into edge-disjoint cycles.

• Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two.

• Application to previous example:

• Four vertices with odd degree!

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Characterization of Eulerian graphs• Theorem. For a connected simple graph G, the following are

equivalent: – G contains an Eulerian circuit.– Every vertex of G has even degree.– The edges of G can be partitioned into edge-disjoint cycles.

• Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two.

• Proof Idea: Eulerian circuits enter and leave vertices on distinct edges so there have to be an even number of edges incident with each vertex. – If you can partition the graph into edge-disjoint cycles, you can string

them together to get circuit.

Characterization of Eulerian graphs• Theorem. For a connected simple graph G, the following are

equivalent: – G contains an Eulerian circuit.– Every vertex of G has even degree.– The edges of G can be partitioned into edge-disjoint cycles.

• Corollary. A connected simple graph has an Eulerian trail if and only if the number of vertices with odd degree is either zero or two.

• Tweaking the example:

• Two vertices with odd degree!

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Visiting “everywhere” on a graph, take 2• A Hamiltonian cycle in a graph is a cycle that visits

every vertex exactly once.

Visiting “everywhere” on a graph, take 2• A Hamiltonian cycle in a graph is a cycle that visits

every vertex exactly once.• Example with an Eulerian trail – does it have a

Hamiltonian cycle?

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Visiting “everywhere” on a graph, take 2• A Hamiltonian cycle in a graph is a cycle that visits

every vertex exactly once.• What about now – does this graph have a

Hamiltonian cycle?

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Visiting “everywhere” on a graph, take 2• A Hamiltonian cycle in a graph is a cycle that visits

every vertex exactly once.• A Hamiltonian graph is a graph that has a

Hamiltonian cycle

Examples

Which of the following is true:A. Both are Hamiltonian and EulerianB. Neither is Hamiltonian and neither is EulerianC. Both are Hamiltonian and not EulerianD. One is Hamiltonian and not Eulerian;

the other is Eulerian and not Hamiltonian E. None of the above.

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Coloring a graph• A proper coloring of a simple graph G = (V,E) is a function λ : V → C, where C is a set of colors, such that{u,v} ∈ E → λ(u) ≠ λ(v).

• There is a proper coloring of a complete graph on 3 vertices using:

A. 1 colorB. 2 colors (but there aren’t any with 1 color)C. 3 colors (but there aren’t any with 1 or 2 colors)D. 4 colors (but there aren’t any with 1 or 2 or 3 colors) E. None of the above.

Another Tree: 3-Coloring a Graph(from Lecture 11)

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A graph is legally colored with k colors if each vertex has a color (label) between 1 and k (inclusive), and no adjacent vertices have the same color.

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Three colors: R, G, BBy symmetry, can use any two colors for vertices 1 and 2

Graph Tree of Colorings

Coloring a graph• A proper coloring of a simple graph G = (V,E) is a function λ : V → C, where C is a set of colors, such that{u,v} ∈ E → λ(u) ≠ λ(v)

• There is a proper coloring of any bipartite graph using:A. 1 colorB. 2 colors (but there aren’t any with 1 color)C. 3 colors (but there aren’t any with 1 or 2 colors)D. 4 colors (but there aren’t any with 1 or 2 or 3 colors) E. None of the above.

Coloring problem and chromatic numbers• The coloring problem: for any c > 2, devise an algorithm whose input can

be any simple graph and whose output is the answer to the question “can the graph be properly colored with c colors?”

• Given G and c, deciding (yes/no) whether G can be properly colored with c colors is an NP-complete problem: no known fast algorithm to come up with solution; but easy to check if candidate solution works.

• The chromatic number of a graph G, (G), is the least number of colors needed to properly color G.

• In the previous example, we saw:(Complete graph, n vertices) = n,

(Bipartite graph) = 2

• The Four Color Theorem says that every planar MAP can be properly colored with four colors. Does this mean that (any planar graph) ≤ 4 ?

– Planar Graph: can be drawn on a plane such that edges intersect only at endpoints

Hard graph problems … we think• Hamiltonian cycle problem: Given a simple graph G, decide

whether G contains a Hamiltonian cycle. • Coloring problem: Given graph G and c > 2, decide whether G

can be properly colored with c colors.• Traveling salesman problem: Given edge-weighted graph G

and a number B, decide whether there is a Hamilton cycle through all vertices of G with total edge cost B.

• Clique problem: Given a simple graph G and a number s, decide whether there are s vertices in G with all C(s,2) possible edges between them being present in G.

• P17.1 Eleven people form five study groups. Prove that two students, A and B, can be found such that every study group that contains A also includes B.

• P17.2 In a game, Player 1 and Player 2 take turns adding to the current number any natural number smaller than it. The game begins with the number 2. Whoever reaches the number 1000 wins. Which player has a winning strategy, and what is the winning strategy?

• P17.3 In a game, Player 1 and Player 2 take turns subtracting from the current number any positive power of 2 less than the current number. The game begins with the number 1000. Whoever reaches the number 0 wins. Which player has a winning strategy, and what is the winning strategy?

Problems 17