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

Why is the base of a logarithm usually left out in Big O, as in O(log n)?

  1. ALogarithms are always rounded to integers
  2. BAll logarithms in computer science have base 10, so it is implied
  3. CLogarithms with different bases differ only by a constant factor
  4. DThe base only matters for small n
2.Algorithms & complexityHard

Taking both endpoints of every edge in a maximal matching gives a vertex cover. How good is this cover?

  1. AAlways a minimum vertex cover
  2. BOf unknown quality compared to the optimum
  3. CAt most log n times the minimum
  4. DAt most twice the size of a minimum vertex cover
3.Data structuresMedium

A binary tree has root 1 with left child 2 and right child 3; node 2 has children 4 (left) and 5 (right). What is the preorder traversal?

  1. A1 2 3 4 5
  2. B4 2 5 1 3
  3. C4 5 2 3 1
  4. D1 2 4 5 3
4.Data structuresHard

In Kruskal's minimum spanning tree algorithm, what is the union-find structure used for?

  1. AStoring each vertex's current distance from the source during the search
  2. BChoosing the starting vertex
  3. CChecking whether an edge's two endpoints are already connected
  4. DSorting the edges by weight
5.Graph theoryMedium

A tree has 4 vertices of degree 3, 2 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?

4 × deg 32 × deg 44 × deg 54 × deg 2
L =
6.Graph theoryHard
1615116292018123ABCDEFG

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 7|E| = 10
W =

3Solutions

1.Algorithms & complexity

Answer: C · Logarithms with different bases differ only by a constant factor

Why this is the answer

Changing the base multiplies the logarithm by a constant, and Big O ignores constant factors.

Answer

C · Logarithms with different bases differ only by a constant factor
2.Algorithms & complexity

Answer: D · At most twice the size of a minimum vertex cover

Why this is the answer

The matching edges share no vertices, so the optimum needs at least one vertex per edge, while this method takes two: a 2-approximation.

Answer

D · At most twice the size of a minimum vertex cover
3.Data structures

Answer: D · 1 2 4 5 3

Why this is the answer

Preorder visits 1, then its left subtree (2, 4, 5), then 3; the other options are inorder, postorder and level-order.

Answer

D · 1 2 4 5 3
4.Data structures

Answer: C · Checking whether an edge's two endpoints are already connected

Why this is the answer

Find tells whether both endpoints are in the same component; if not, the edge is added and union merges the components.

Answer

C · Checking whether an edge's two endpoints are already connected
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 + 48 = 2(L + 14) − 2
L = 22

Answer

L = 22

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

6.Graph theory

Answer: W = 47

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.

Kruskal’s algorithm

DE (2): take
GE (3): take
CE (6): take
DF (9): take
BD (11): take
GD (12): skip, closes a cycle
BC (15): skip, closes a cycle
AB (16): take

Total weight

W = 2 + 3 + 6 + 9 + 11 + 16 = 47

Answer

W = 47

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