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 the recurrence T(n) = T(n − 1) + O(n) solve to?

  1. AO(n²)
  2. BO(n log n)
  3. CO(2ⁿ)
  4. DO(n)
2.Algorithms & complexityHard

Which definition of the class NP is correct?

  1. ADecision problems whose yes-answers can be verified in polynomial time given a certificate
  2. BProblems that are not polynomial
  3. CProblems that cannot be solved in polynomial time
  4. DProblems that no algorithm can solve
3.Data structuresMedium

Which data structure does a running program use to keep track of active function calls and their local variables?

  1. AA binary search tree of frames
  2. BThe call stack
  3. CA hash table of functions
  4. DThe heap memory queue
4.Data structuresHard

What is the main advantage of a skip list over a sorted linked list?

  1. AGuaranteed O(1) search because every node stores a pointer to every other node
  2. BIt uses no extra memory
  3. CIt stores keys without order
  4. DExpected O(log n) search by using several levels of forward pointers
5.Graph theoryMedium
ABCDEFEuler 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| = 6|E| = 6
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
13913731261313ABCDEFG

Use Dijkstra’s algorithm to find the length d(G) of the shortest path from A to G in the weighted graph.

start: Aend: G
d(G) =

3Solutions

1.Algorithms & complexity

Answer: A · O(n²)

Why this is the answer

The work sums to about n² / 2, which is the recurrence of selection sort or of quicksort with the worst pivot.

Answer

A · O(n²)
2.Algorithms & complexity

Answer: A · Decision problems whose yes-answers can be verified in polynomial time given a certificate

Why this is the answer

For an NP problem, a proposed solution (certificate) can be checked quickly, for example verifying a given assignment satisfies a Boolean formula.

Answer

A · Decision problems whose yes-answers can be verified in polynomial time given a certificate
3.Data structures

Answer: B · The call stack

Why this is the answer

Each call pushes a stack frame and each return pops it, which is why the most recent call finishes first.

Answer

B · The call stack
4.Data structures

Answer: D · Expected O(log n) search by using several levels of forward pointers

Why this is the answer

Randomised levels allow searches to skip many nodes, giving expected O(log n) search, insert and delete.

Answer

D · Expected O(log n) search by using several levels of forward pointers
5.Graph theory

Answer: Euler path, no 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) = 3, deg(C) = 2, deg(D) = 2, deg(E) = 2, deg(F) = 1

Odd vertices

B, F → 2
Euler path, no circuit

Answer

Euler path, no circuit

The Euler path starts at one odd vertex and ends at the other.

6.Graph theory

Answer: d(G) = 23

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.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): B: 13, C: 9
C (9): B: 10, E: 16
B (10): D: 13, G: 23
D (13): F: 25
E (16): F: 22
F (22)
G (23)

Shortest path

A → C → B → G
d(G) = 23

Answer

d(G) = 23

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