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1 Clicker COMPSC 311 Section 1: ntroduction to Algorithms Lecture 3: Asymptotic Notation and Efficiency + Graph ntro Dan Sheldon University of Massachusetts Suppose f is O(g). Which of the following is true? A. g is O(f) B. g is not O(f) C. g may be O(f), depending on the particular functions f and g {February 4, 2019} Big-Ω Motivation More Big-Ω Motivation Algorithm foo do something... Fact: run time is O(n 3 ) Algorithm bar for k= 1 to n do do something else.. Fact: run time is O(n 3 ) Algorithm sum-product sum = 0 for j= i to n do sum += A[i]*A[j] What is the running time of sum-product? Conclusion: foo and bar have the same asymptotic running time. What is wrong? Easy to see it is O(n 2 ). Could it be better? O(n)? Big-Ω Clicker nformally: T grows at least as fast as f Definition: The function T (n) is Ω(f(n)) if there exist constants c > 0 and n 0 0 such that T (n) cf(n) for all n n 0 f is an asymptotic lower bound for T Which is an equivalent definition of big Omega notation? A. T (n) is Ω(f(n)) if f(n) is O(T (n)) B. T (n) is Ω(f(n)) if there exists a constant c > 0 such that T (n) c f(n) for infinitely many n C. Both A and B D. Neither A nor B

2 Big-Ω Solution Exercise: let T (n) be the running time of sum-product. Show that T (n) is Ω(n 2 ) Algorithm sum-product sum = 0 for j= i to n do sum += A[i]*A[j] Hard way Count exactly how many times the loop executes Easy way n = n(n + 1) 2 = Ω(n 2 ) gnore all loop executions where i > n/2 or j < n/2 The inner statement executes at least (n/2) 2 = Ω(n 2 ) times Big-Θ Big-Θ example How do we correctly compare the running time of these algorithms? Definition: the function T (n) is Θ(f(n)) if it is both O(f(n)) and Ω(f(n)). f is an asymptotically tight bound of T Example. T (n) = 32n n + 1 T (n) is Θ(n 2 ) T (n) is neither Θ(n) nor Θ(n 3 ) Algorithm foo do something... Algorithm bar for k= 1 to n do do something else.. Answer: foo is Θ(n 2 ) and bar is Θ(n 3 ). They do not have the same asymptotic running time. Additivity Revisited Running Time Analysis Suppose f and g are two (non-negative) functions and f is O(g) Old version: Then f + g is O(g) New version: Then f + g is Θ(g) }{{} n n + n log n is Θ(n 2 ) }{{} g f Mathematical analysis of worst-case running time of an algorithm as function of input size. Why these choices? Mathematical: describes the algorithm. Avoids hard-to-control experimental factors (CPU, programming language, quality of implementation), while still being predictive. Worst-case: just works. ( average case appealing, but hard to analyze) Function of input size: allows predictions. What will happen on a new input?

3 Efficiency Polynomial Time When is an algorithm efficient? Stable Matching Brute force: Ω(n!) Propose-and-Reject?: O(n 2 ) We must have done something clever Definition: an algorithm runs in polynomial time if its running time is O(n d ) for some constant d Polynomial Time: Examples Why Polynomial Time? These are polynomial time: f 1 (n) = n f 2 (n) = 4n f 3 (n) = n log(n) + 2n + 20 f 4 (n) = 0.01n 2 f 5 (n) = n 2 f 6 (n) = 20n 2 + 2n + 3 Not polynomial time: f 7 (n) = 2 n f 8 (n) = 3 n f 9 (n) = n! Why is this a good definition of efficiency? Matches practice: almost all practically efficient algorithms have this property. Usually distinguishes a clever algorithm from a brute force approach. Refutable: gives us a way of saying an algorithm is not efficient, or that no efficient algorithm exists. Graphs: Motivation Networks Shortest driving route from Amherst to Florida? Number of degrees of separation between you and Tom Brady (or Theresa May) in online social network? Find influencers and bots on twitter? Find reputable web pages? How do we build algorithms to answer these questions? Graphs and graph algorithms.

4 One week of Enron s Political blogosphere graph Node = political blog; edge = link. The Political Blogosphere and the 2004 U.S. Election: Divided They Blog, Adamic and Glance, slide credit: Kevin Wayne / Pearson Applications Graphs longer existed, or had moved to a different location. When looking at the front page of a blog we did Networks (real, online, etc.) Shortest driving route from Amherst to Florida Number of degrees of separation between you and Tom Brady nfluencers / bots on twitter Reputable pages on web + many more mage segmentation Airplane scheduling Program analysis: control flow, function calls Playing chess (A search) + many more A graph is a mathematical representation of a network Set of nodes (vertices) V Set of pairs of nodes (edges) E Notation: n = V, m = E (almost always) 4 Example: nternet in PATHS AND CONNECTVTY not make a distinction between blog references made in blogrolls (blogroll links) from those made in posts (post citations). This had the disadvantage of not differentiating between blogs that were actively mentioned in a post on that day, from blogroll links that remain static over many weeks [10]. Since posts usually contain sparse references to other blogs, and blogrolls usually contain dozens of blogs, we assumed that the network obtained by crawling the front page of each blog would strongly reflect blogroll links. 479 blogs had blogrolls through blogrolling.com, while many others simply maintained a list of links to their favorite blogs. We did not include blogrolls placed on a secondary page. We constructed a citation network by identifying whether a URL present on the page of one blog references another political blog. We called a link found anywhere on a blog s page, a page link to distinguish it from a post citation, a link to another blog that occurs strictly within a post. Figure 1 the unmistakable division between the liberal and conservative political (blogo)spheres. n shows fact, 91% of the links originating within either the conservative or liberal communities stay within thatcommunity. An effect that may not be as apparent from the visualization is that even though we started with a balanced set of blogs, conservative blogs show a greater tendency to link. 84% of conservative blogs link to at least one other blog, and 82% receive a link. n contrast, 74% of liberal blogs link to another blog, while only 67% are linked to by another blog. So overall, we see a slightly higher tendency for conservative blogs to link. Liberal blogs linked to 13.6 blogs on average, while conservative blogs linked to an average of 15.1, and this difference is almost entirely due to the higher proportion of liberal blogs with no links at all. Although liberal blogs may not link as generously on average, the most popular liberal blogs, Daily Kos and Eschaton (atrios.blogspot.com), had 338 and 264 links from our single-day snapshot Graph G = (V, E) Basic building block of many other algorithms / analyses Figure 1: Community structure of political blogs (expanded set), shown using utilizing a GEM layout [11] in the GUESS[3] visualization and analysis tool. The colors reflect political orientation, red for conservative, and blue for liberal. Orange links go from liberal to conservative, and purple slide credit: Kevin Wayne / Pearson ones from conservative to liberal. The size of each blog reflects the number of other blogs that link to it. Example: nternet in CHAPTER 2. GRAPHS SR UCSB STAN UCLA LNC MT BBN CARN SDC RAND HARV Figure 2.3: An alternate drawing of the 13-node nternet graph from December Edge e =Although {u, v}we ve but usuallyexamples written e = (u, v) di erent areas, there Paths. been discussing of graphs in many are clearly common themes in the use ofendpoints graphs across of these u and v aresome neighbors, adjacent, e areas. Perhaps foremost these is the idea that things often travel across the edges of a graph, moving from e isamong incident to u and v Figure 2.2: A network depicting the sites on the nternet, then known as the Arpanet, in December (mage from F. Heart, A. McKenzie, J. McQuillian, and D. Walden [214]; on-line at connections such as hyperlinks, citations, or cross-references. The list of areas in which graphs play a role is of course much broader than what we can enumerate here; Figure 2.4 gives a few further examples, and also shows that many images we encounter on a regular basis have graphs embedded in them. ucla is a path in the nternet graph from Figures 2.2 and 2.3, as is the sequence case, edges, in which the first and last nodes are the same, but otherwise all nodes are distinct. 37

5 Example: nternet in CHAPTER 2. GRAPHS Example: nternet in CHAPTER 2. GRAPHS SR LNC SR LNC MT MT UCSB STAN SDC UCSB STAN SDC BBN CARN BBN CARN UCLA RAND HARV UCLA RAND HARV Figure 2.3: An alternate drawing of the 13-node nternet graph from December A path is a sequence P = v 1, v 2,..., v k 1, v k such that each consecutive pair v i, v i+1 is joined by an edge in G ucla is a path in the nternet graph from Figures 2.2 and 2.3, as is the sequence case, Path from v 1 to v k. A v 1 v k path Example: nternet in Cycles. A particularly important kind of non-simple path is a cycle, CHAPTER which informally 2. GRAPHS is a SR LNC edges, in which the first and last nodes are the same, but otherwise all nodes are distinct. There are many cycles in Figure 2.3: sri, stan, ucla, sri is as short an example as possible MT according to our definition (since it has exactly three edges), while sri, stan, ucla, rand, bbn, mit, UCSButah, sri isstan a significantlysdc longer example. n fact, every edge in the 1970 Arpanet belongs to a cycle, BBN and this was by design: CARN it means that if any edge were to fail (e.g. a construction crew accidentally cut through the cable), there would stillucla be a way to RAND get from any node to any other node. HARV More generally, cycles Figure 2.3: An alternate drawing of the 13-node nternet graph from December Simple path, cycle, distance ucla is a path in the nternet graph from Figures 2.2 and 2.3, as is the sequence case, example, 26 sri, stan, ucla, sri, utah, mit is a path. But most paths CHAPTER we consider 2. GRAPHS will not Example: nternet in 1970 MT on the UCSB right-hand-side of STAN Figure 2.3. SDC More precisely, a cycle is a path with at least three edges, in which the first and last nodes are the same, butbbn otherwise all nodes are CARN distinct. There are many cycles in Figure 2.3: sri, stan, ucla, sri is as short an example as possible according to ourucla definition (since RANDit has exactly three edges), while sri, HARV stan, ucla, rand, bbn, mit, utah, sri is a significantly longer example. n fact, every edge in the 1970 Arpanet belongs to a cycle, and this was by design: it means that Figure if any2.3: edge Anwere alternate to fail drawing (e.g. a of construction the 13-nodecrew nternet accidentally graph from cut through December the cable), there would still be a way to get from any node to any other node. More generally, cycles SR Connected graph = graph with paths between every pair of vertices. ucla is a path in the nternet graph from Figures 2.2 and 2.3, as is the sequence case, Connected component? LNC Figure 2.3: An alternate drawing of the 13-node nternet graph from December Q: Which is not a path? ucla is a path in the nternet graph from Figures 2.2 and 2.3, as is the sequence case, A. UCSB - SR - B. LNC - MT - LNC - C. UCSB - SR - STAN - UCLA - UCSB D. None of the above Definitions edges, in which the first and last nodes are the same, but otherwise all nodes are distinct. Simple path: path where all vertices are distinct There are many cycles in Figure 2.3: sri, stan, ucla, sri is as short an example as possible according to our definition (since it has exactly three edges), while sri, stan, ucla, rand, (Simple) Cycle: path v bbn, mit, utah, sri is a significantly 1,..., v longer example. k 1, v k where n fact, every edge in the 1970 Arpanet belongs to a cycle, and this was by design: it means v1 = v k that if any edge were to fail (e.g. a construction crew accidentally cut through the cable), there would First still k be a way 1 nodes to get from distinct any node to any other node. More generally, cycles All edges distinct (k > 3) Distance from u to v: minimum number of edges in a u v path Definitions Connected component: maximal subset of nodes such that a path exists between each pair in the set maximal = if a new node is added to the set, there will no longer be a path between each pair edges, in which the first and last nodes are the same, but otherwise all nodes are distinct.

6 Definitions Directed Graphs Tree: a connected graph with no cycles Q: s this equivalent to trees you saw in Data Structures? A: More or less. Graphs can be directed, which means that edges point from one node to another, to encode an asymmetric relationship. We ll talk more about directed graphs later. Graphs are undirected if not otherwise specified. Rooted tree: tree with parent-child relationship Pick root r and orient all edges away from root Parent of v = predecessor on path from r to v Graph Traversal Thought experiment. World social graph. s it connected? f not, how big is largest connected component? s there a path between you and Tom Brady? What about Theresa May? How can you tell algorithmically? Answer: graph traversal! (BFS/DFS)