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

How does Prim's algorithm build a minimum spanning tree?

  1. AIt sorts all edges by weight and adds each one unless it would form a cycle in the forest
  2. BIt grows one tree from a start vertex, adding the cheapest edge that leaves the tree
  3. CIt removes the most expensive edge until no cycles remain
  4. DIt runs breadth-first search and keeps the BFS tree
2.Algorithms & complexityHard

Using the master theorem, T(n) = 8T(n/2) + O(n²) solves to:

  1. AΘ(n log n)
  2. BΘ(n²)
  3. CΘ(n² log n)
  4. DΘ(n³)
3.Data structuresMedium

Why is traversing an array often much faster in practice than traversing a linked list of the same length?

  1. AArray traversal is O(1) in total, while linked list traversal is O(n)
  2. BLinked lists always store elements on disk
  3. CArrays use fewer instructions to compare values
  4. DArray elements are contiguous, so they make good use of the CPU cache
4.Data structuresHard

Why are B-trees used in databases and file systems?

  1. ATheir high branching factor keeps the tree shallow, reducing slow disk reads
  2. BThey use less memory than any array, so the entire index always fits in main memory
  3. CThey store data unsorted
  4. DThey need no rebalancing
5.Graph theoryMedium

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

3 × deg 32 × deg 42 × deg 5
L =
6.Graph theoryHard
ABCDE

How many walks of length 2 go from D to C? A walk may repeat vertices and edges. Call the number N.

M = adjacency matrixL = 2
N =

3Solutions

1.Algorithms & complexity

Answer: B · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree

Why this is the answer

Prim's algorithm, with a priority queue of crossing edges, runs in O(E log V) with a binary heap.

Answer

B · It grows one tree from a start vertex, adding the cheapest edge that leaves the tree
2.Algorithms & complexity

Answer: D · Θ(n³)

Why this is the answer

log₂ 8 = 3, and n² grows slower than n³, so the leaves dominate: Θ(n³).

Answer

D · Θ(n³)
3.Data structures

Answer: D · Array elements are contiguous, so they make good use of the CPU cache

Why this is the answer

Contiguous memory gives good locality of reference, while linked nodes may be scattered and cause many cache misses.

Answer

D · Array elements are contiguous, so they make good use of the CPU cache
4.Data structures

Answer: A · Their high branching factor keeps the tree shallow, reducing slow disk reads

Why this is the answer

B-trees store many keys per node, so height is small and few block reads are needed per lookup; B+ trees also link leaves for range scans.

Answer

A · Their high branching factor keeps the tree shallow, reducing slow disk reads
5.Graph theory

Answer: L = 15

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 + 27 = 2(L + 7) − 2
L = 15

Answer

L = 15

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

6.Graph theory

Answer: N = 2

Degrees: the degree of a vertex is the number of edges that meet it. Every edge has two ends, so the degrees add up to twice the number of edges: Σ deg(v) = 2|E| (the handshake lemma). The number of vertices with odd degree is therefore even. Kn has n(n − 1)/2 edges, a k-regular graph on n vertices has nk/2, and a tree on n vertices has n − 1 edges.

Adjacency matrix

A: 0 1 1 0 0
B: 1 0 1 1 0
C: 1 1 0 0 1
D: 0 1 0 0 1
E: 0 0 1 1 0

Common neighbours

(M²)DC = 0·1 + 1·1 + 0·0 + 0·0 + 1·1 = 2

Answer

N = 2

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