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.

New problemsBack to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

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

In a sorted array, you look for two elements that sum to a target by moving one pointer from each end inwards. What is the time complexity?

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

Strassen's algorithm multiplies matrices using 7 instead of 8 recursive multiplications of half-size blocks. What is its running time?

  1. AΘ(nlog₂ 7), about n2.81
  2. BΘ(n2)
  3. CΘ(n³)
  4. DΘ(n² log n)
3.Data structuresMedium

What is the value of the postfix expression 3 4 + 2 * ?

  1. A24
  2. B10
  3. C11
  4. D14
4.Data structuresHard

Which property keeps every B-tree perfectly balanced?

  1. AEach node stores one key
  2. BAll leaves are at the same depth
  3. CThe keys are stored only in the root
  4. DEvery node has exactly two children
5.Graph theoryMedium

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

3 × deg 42 × deg 2
L =
6.Graph theoryHard
19131353111411ABCDEF

Dijkstra’s algorithm runs from A. Click the vertex that is settled (made permanent) in round 5; A itself is round 1.

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

3Solutions

1.Algorithms & complexity

Answer: D · O(n)

Why this is the answer

If the sum is too small the left pointer moves right, if too large the right pointer moves left, so there are at most n steps.

Answer

D · O(n)
2.Algorithms & complexity

Answer: A · Θ(nlog₂ 7), about n2.81

Why this is the answer

With a = 7 and b = 2, nlog₂ 7 dominates the O(n²) addition work, giving about n2.81.

Answer

A · Θ(nlog₂ 7), about n2.81
3.Data structures

Answer: D · 14

Why this is the answer

Push 3 and 4, + gives 7; push 2, * gives 7 × 2 = 14.

Answer

D · 14
4.Data structures

Answer: B · All leaves are at the same depth

Why this is the answer

Nodes split upwards when full and the tree only gets taller when the root splits, so all leaves always stay on the same level.

Answer

B · All leaves are at the same depth
5.Graph theory

Answer: L = 8

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 + 16 = 2(L + 5) − 2
L = 8

Answer

L = 8

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

6.Graph theory

Answer: D

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: 1, C: 9
2. B (1): D: 14, E: 12
3. C (9)
4. E (12): F: 26
5. D (14): F: 25
6. F (25)

round 5

D

Answer

D

ExamProblems · examproblems.com/courses/data-structures-and-algorithms