CS 173: Skills list for Examlet 5

• Trees
  • Define a (full) binary tree recursively: it's either a single node, or a root with two subtrees children, or (for non-full trees) a root with one subtree child.
  • Define and use tree terminology: root, leaf, internal node, parent, child, sibling, (proper) ancestor, (proper) descendent, level of a node, height of tree.
  • Define what it means for a tree to be binary, n-ary, or balanced. (The definitions for "full" and "complete" trees will be provided.)
  • Know key facts: a tree has one more node than edges, a full m-ary tree with i internal nodes has mi+1 nodes total, a full binary tree with n internal nodes has n+1 leaves (and thus 2n+1 nodes total),
  • Know key facts about binary trees of height h: the number of leaves is between 1 and 2^h, the number of nodes is between (h+1) and (2^h+1 - 1)
  • Know that the height of a full complete binary tree with n nodes (or n leaves) is approximately log(n).
• String notation and regular expressions
  • The empty string (ε)
  • Concatenation, *, and | operations
  • A^* is the set of all finite-length strings with characters from A.
  • Know what a "bit string" is (a string of ones and zeroes)
• Grammar Trees
  • Correctly interpret the definition of a context-free grammar: variables, rules, start symbols, terminal symbols
  • Correctly interpret a grammar rule, e.g. what does ε or a vertical bar mean on the righthand side of a rule.
  • Given a grammar G, give examples of trees that are/aren't generated by G, determine whether a given tree could be generated by G.
  • Given a string s and a grammar G, briefly explain why s can/can't be generated by G.
  • Build a tree matching grammar G with a specific terminal string s. When multiple trees are possible, build more than one or describe the set of possible trees.
• Tree induction
  • Prove a claim about labelled or unlabelled trees using induction.
  • Use induction to prove a claim about other sorts of objects, where your induction variable is the size of the object and there may be more than one object of each size.
• Big O
  • Define what it means for a function f to be asymptotically smaller than g (f ≪ g), O(g) and/or θ(g). where g is another function.
  • For specific functions f and g, identify whether f is asymptotically smaller than g (f ≪ g), O(g) and/or θ(g).
  • Know the asymptotic relationships among key primitive functions: constant, log n, n, n log n, polynomials of higher orders, exponentials, factorial.
  • Given functions f and g, prove that f is O(g) and/or θ(g).
• Algorithms
  • Know the big-O running times of basic operations on linked lists and arrays. These include reading/writing values, adding/removing elements, and dividing the list/array at locations at/near the start, in the middle, and at the end.
  • Be familiar with the overall structure and big-O running times of the following algorithms.
    • merge two sorted lists
    • binary search
    • merge sort
    • graph reachability (what's in x's connected component)