1. Final Review Chris and Virginia

2. Overview One big multi-part question. (Likely to be on data structures) Many small questions. (Similar to those in midterm 2) A few questions from the homework and class. (This slide is on the class webpage)

3. You need to able to Summarize what we learned. Recognize false proofs. Use appropriate data structures. Invent new data structures. Design new (e.g. randomized) algorithms. Know NP, coNP, NP-completeness, approximation algorithms Matching, Graphs, and random walk.

4. Summarize what we learned. Given a problem that appeared in class or homework, –Describe the approach to solve the problem –Summarize the answer in 100 words or less. You do NOT need to remember the whole answer. We will NOT ask you to repeat the entire solution.

5. Example: Planar 5 coloring

6. 1)Find v, any vertex of degree 5 or less. –Always possible, since |E| < 6 |V| in planar graph. 2)Merge x, y such that (v, x) in E, (v, y) in E, but (x, y) not in E –We can always find x, y, since planar graph has no 5-clique. 3)Merge x and y, and remove v. –v will has only 4 neighbors, so there always exists a color for v 4)Repeat Step 1 until there are no vertices 5)Color the vertices in reverse order of removal.

7. Recognize false proofs/argument “Proof”: The above algorithm clearly runs in O(n).

8. Recognize false proofs/argument “Proof”: The above algorithm clearly runs in O(n). Problem in step 2: “Find x, y such that (v, x) in E, (v, y) in E, but (x, y) not in E.” –we don’t have adj. matrix to query in O(1) –we can not afford to construct adj. matrix since we only have O(n) total –(Btw, this is not good answer, and Manuel would not let me put something ambiguous on the final.)

9. What data structure to use? You need Insert(x), Delete(x) and Find(x), and you query a small subset of all the elements very frequently.

10. What data structure to use? You need Insert(x), Delete(x) and Find(x), and you query a small subset of all the elements very frequently. –Splay tree, because splay tree moves recently accessed node to the top, so it is cheaper to access it again.

11. What data structure to use? You need Insert(x), Delete(x) and Findmin(x), and Findmax(x).

12. What data structure to use? You need Insert(x), Delete(x) and Findmin(x), and Findmax(x). –If a pointer to x is given for delete Use a maxheap and a minheap. –If not, Use a binary search tree and maintain the pointers to min and max elements.

13. What data structure to use? You need Insert(x), Delete(x) (given pointer to x) and FindMedian().

14. What data structure to use? You need Insert(x), Delete(x) (given pointer to x) and FindMedian(). –Use two heaps. A maxheap to store everything less than the median, and a minheap to store everything greater than the median.

15. Prove lower bound Can you design a data structure where Insert(x), Delete(x) and FindMedian() are all little o(log n)?

16. Prove lower bound Can you design a data structure where Insert(x), Delete(x) and FindMedian() are all little o(log n)? –No. –Sorting lower bounds: Any algorithm that sorts the set of elements, S, must perform at least Omega(|S| log |S|) comparisons between elements of S.

17. Example: MINSAT Given a formula in CNF and an integer K, is there an assignment which satisfies at most K clauses. (no restriction on clause size, except that the literals are distinct; no duplicate clauses) Decision problems, NP and coNP NP-hardness and completeness

18. MINSAT cont. Reduction from Vertex Cover (VC). Given [G=(V,E), K] instance of VC. Direct the edges arbitrarily. Create a clause C u for each vertex u For (u,v) add x uv to C u and ¬x uv to C v.

19. Approximation Algorithms Approximation Ratio: M alg /M opt Example: 2-approximation for MINSAT Convert to VC: –A vertex for each clause –Edge between clauses if they have conflicting literals: x and ¬x. –Approximate VC instance.

20. Planarity You should know: –Planar graph definition, duals, properties –Planarity testing/planar embedding in linear time –Planar Separator Theorem: finding a small separator in linear time

21. Planarity Example: a (1/3,2/3)-separator of size 2 for outerplanar graphs A graph is outerplanar if it has a plane embedding so that all vertices are on the outer face

22. Outerplanar Separator Given the outerplanar embedding of G, triangulate all inner faces. Create the plane dual, except for the vertex of the outer face; this is a TREE of degree at most 3. Find (1/3,2/3)-edge separator of the tree. Remove the two vertices of the corresponding edge in G – this is the separator.