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Final Review

Final Review. Chris and Virginia. 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). You need to able to.

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Final Review

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  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. Example: Planar 5 coloring • Find v, any vertex of degree 5 or less. • Always possible, since |E| < 6 |V| in planar graph. • 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. • Merge x and y, and remove v. • v will has only 4 neighbors, so there always exists a color for v • Repeat Step 1 until there are no vertices • 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. Decision problems, NP and coNPNP-hardness and completeness • Example: MINSAT Given a formula in CNF and an integer K, is there an assignment which satisfies at mostK clauses. (no restriction on clause size, except that the literals are distinct; no duplicate clauses)

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

  19. Approximation Algorithms • Approximation Ratio: Malg/Mopt • 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.

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