Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

What does it mean that a problem has optimal substructure?

  1. AThe problem can be solved in constant time
  2. BEvery subproblem has the same size
  3. CThe problem has exactly one optimal solution, which every subproblem shares
  4. DAn optimal solution can be built from optimal solutions of its subproblems
2.Algorithms & complexityHard

To show that a new problem X is NP-hard, what must you do?

  1. AReduce a known NP-hard problem to X in polynomial time
  2. BShow that X has no polynomial algorithm for small inputs
  3. CShow that X can be verified in polynomial time
  4. DReduce X to a known NP-hard problem in polynomial time
3.Data structuresMedium

A queue is implemented with two stacks: push onto an "in" stack, and pop from an "out" stack, refilling it from "in" when empty. What is the amortised cost per operation?

  1. AO(n)
  2. BO(log n)
  3. CO(n²)
  4. DO(1)
4.Data structuresHard

Which structure lets you compute the maximum of every window of size k in an array in O(n) total?

  1. AA sorted array updated by shifting
  2. BA singly linked list of the window's elements
  3. CA deque holding indices in decreasing order of value
  4. DA min-heap rebuilt for every window
5.Graph theoryMedium
ABCDEEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 5|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
5131681918171310ABCDEFG

Prim’s algorithm builds a minimum spanning tree starting from vertex E. Click the edge it adds as edge number 2.

start: Ek = 2
  1. 1edge AB
  2. 2edge BC
  3. 3edge BD
  4. 4edge CE
  5. 5edge DE
  6. 6edge DF
  7. 7edge EF
  8. 8edge GB
  9. 9edge GC
  10. 10edge GE

3Solutions

1.Algorithms & complexity

Answer: D · An optimal solution can be built from optimal solutions of its subproblems

Why this is the answer

Optimal substructure is required for both greedy algorithms and dynamic programming to produce correct answers.

Answer

D · An optimal solution can be built from optimal solutions of its subproblems
2.Algorithms & complexity

Answer: A · Reduce a known NP-hard problem to X in polynomial time

Why this is the answer

If a known hard problem can be transformed into X efficiently, a fast algorithm for X would solve the known problem too.

Answer

A · Reduce a known NP-hard problem to X in polynomial time
3.Data structures

Answer: D · O(1)

Why this is the answer

Each element is pushed and popped at most twice in total, so although one refill can take O(n), the average over a sequence is O(1).

Answer

D · O(1)
4.Data structures

Answer: C · A deque holding indices in decreasing order of value

Why this is the answer

The front of the deque is always the window's maximum; each index enters and leaves the deque once, so the work is linear.

Answer

C · A deque holding indices in decreasing order of value
5.Graph theory

Answer: Euler circuit

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Degrees

deg(A) = 2, deg(B) = 4, deg(C) = 2, deg(D) = 2, deg(E) = 2

Odd vertices

0
Euler circuit

Answer

Euler circuit
6.Graph theory

Answer: edge GC

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Prim’s algorithm

1. CE (8)
2. GC (3)
3. GB (1)
4. AB (5)
5. BD (16)
6. EF (17)

edge number 2

GC

Answer

edge GC

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