# MidtermReview

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Basic algorithmic concepts --- Lecture1, Lecture2, Lecture3
• correctness of an algorithm
• worst-case running time as a function of input size
• example: Euclid's algorithm

Mathematics --- Lecture3, Lecture4

• O-notation, Θ(), Ω()
• How do you tell whether a sum is a geometric sum?
• Is a geometric sum proportional to its largest term?
• Is this a geometric sum: i=1..n i2?
• Is that sum proportional to its largest term?
• Give the best big-O upper bound you can on i=1..n i log i.
• Give the best big-Θ lower bound you can on i=1..n i log i.
• Describe the recursion trees for the following recurrences:
T(n) = 3T(n-3); T(0) = 1;
S(n) = 3S(n/3); T(0) = 1;
For each tree, what is the depth and how many children does each node have?
Give the best O and Θ bounds you can on T(n) and S(n).
• Draw a directed acyclic graph with 10 vertices, choose a source vertex, and label each vertex with the number of paths from the source to that vertex.
• Describe a linear-time algorithm for doing this in arbitrary directed acyclic graphs.
• Describe the recursion tree for the following algorithm:
1. int fib(n) { if (n<= 1) return n; return f(n-1)+f(n-2); }
Argue that the depth of the tree is at least n/2 and at most n.
Argue that the running time of the algorithm is at least 2n/2.
• Describe an algorithm running in O(n) time for computing the n'th fibonacci number.
• Argue that it runs in O(n) time.
• Define "n choose k" = C(n,k).
• Give a recurrence relation for C(n,k).
• Describe an algorithm running in O(nk) time for computing C(n,k).
• Define the subset sum problem.
• Describe a dynamic programming algorithm for the problem.
• What is the running time?
• What is the underlying recurrence relation?

Longest ascending subsequence, Longest common subsequence (book section 11.5)

• Know the following terms: neighbor, path, cycle, tree, connected graph, connected component, vertex degree.
• By hand, run DFS on some undirected and directed graphs. Show the resulting DFS tree and the DFS numbering.
• In an undirected graph, explain why DFS does not classify any edges as cross edges or forward edges.
• What is the worst-case time complexity of DFS on a graph with n nodes and m edges?
• Justify your answer. Give a clear argument bounding the time taken by DFS in terms of n and m.
• What is the definition of a cut vertex?
• Define what the low numbers are, in the algorithm for finding cut vertices.
• How can you tell whether a vertex is a cut vertex by looking at the low numbers?
• Give a recurrence relation for the low numbers.
• Explain how to use that recurrence relation to find cut vertices in linear time.
• Prove that a directed graph has a cycle if and only if DFS will classify some edge of the graph as a back edge.
• Define topological ordering of a directed acyclic graph.
• Give an example.
• If you order the vertices by DFS number, does that always give a topological ordering?
• Define the DFS post-order numbering.
• Give pseudo-code to compute the DFS post-order numbering.
• If you order the vertices by DFS post-order number, does that always give a topological ordering?

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