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

Why do many library sorting routines switch to insertion sort for very small subarrays?

  1. AInsertion sort has little overhead and is fast on small or nearly sorted inputs
  2. BRecursive algorithms cannot handle arrays below 10 elements
  3. CInsertion sort uses less memory than any other sorting algorithm on large inputs
  4. DInsertion sort is O(n log n) in the worst case
2.Algorithms & complexityHard

Which definition of the class NP is correct?

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

How does extract-min work in a binary min-heap?

  1. ARemove the last element of the array
  2. BSearch all leaves for the new minimum
  3. CMove the last element to the root and sift it down
  4. DRemove the root and shift all elements one place left
4.Data structuresHard

What is the running time of breadth-first search on a graph with V vertices stored as an adjacency matrix?

  1. AO(E log V)
  2. BO(V)
  3. CO(V²)
  4. DO(V + E)
5.Graph theoryMedium

A tree has 2 vertices of degree 3, 3 vertices of degree 4, 4 vertices of degree 5, 4 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

2 × deg 33 × deg 44 × deg 54 × deg 2
L =
6.Graph theoryHard
15144311536ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 3; A itself is round 1.

start: Around 3
  1. 1A
  2. 2B
  3. 3C
  4. 4D
  5. 5E
  6. 6F

3Solutions

1.Algorithms & complexity

Answer: A · Insertion sort has little overhead and is fast on small or nearly sorted inputs

Why this is the answer

For small n the simple loop of insertion sort beats the recursion overhead of quicksort or merge sort, even though it is O(n²).

Answer

A · Insertion sort has little overhead and is fast on small or nearly sorted inputs
2.Algorithms & complexity

Answer: B · 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

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

Answer: C · Move the last element to the root and sift it down

Why this is the answer

The last element fills the hole at the root and is swapped with its smaller child until the heap property holds, taking O(log n).

Answer

C · Move the last element to the root and sift it down
4.Data structures

Answer: C · O(V²)

Why this is the answer

Each of the V vertices scans a row of V entries, giving O(V²); with adjacency lists BFS is O(V + E).

Answer

C · O(V²)
5.Graph theory

Answer: L = 22

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.

Handshake lemma for a tree

L + 46 = 2(L + 13) − 2
L = 22

Answer

L = 22

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: B

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

1. A (0): B: 15, C: 14
2. C (14): E: 25
3. B (15): D: 18, E: 21
4. D (18): F: 23
5. E (21)
6. F (23)

round 3

B

Answer

B

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