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

The lower bound for comparison-based sorting in the worst case is:

  1. AΩ(n²)
  2. BΩ(n log n)
  3. CΩ(n)
  4. DΩ(log n)
2.Algorithms & complexityHard

What would follow if someone found a polynomial-time algorithm for one NP-complete problem?

  1. AThe halting problem would become decidable
  2. BEvery problem in NP could be solved in polynomial time, so P = NP
  3. CAll exponential-time algorithms would immediately run in linear time
  4. DOnly that one problem would move into P
3.Data structuresMedium

Which data structure is used by depth-first search (iterative version)?

  1. AA heap
  2. BA queue
  3. CA stack
  4. DA hash table only
4.Data structuresHard

Why do databases often use B+ trees rather than plain B-trees for indexes?

  1. AB+ trees use binary internal nodes, which reduces the number of disk reads
  2. BB+ trees do not need to be balanced
  3. CAll records are in linked leaves, which makes range scans efficient
  4. DB+ trees store data only in the root
5.Graph theoryMedium

A connected graph has 6 vertices and 15 edges. How many edges k must be removed to leave a spanning tree?

|V| = 6|E| = 15
k =
6.Graph theoryHard
5128913717203126ABCDEFG

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

|V| = 7|E| = 12
W =

3Solutions

1.Algorithms & complexity

Answer: B · Ω(n log n)

Why this is the answer

Any comparison sort needs at least log₂(n!) ≈ n log n comparisons in the worst case. Counting sort beats it because it does not compare.

Answer

B · Ω(n log n)
2.Algorithms & complexity

Answer: B · Every problem in NP could be solved in polynomial time, so P = NP

Why this is the answer

Since all of NP reduces to any NP-complete problem in polynomial time, a fast algorithm for one gives a fast algorithm for all.

Answer

B · Every problem in NP could be solved in polynomial time, so P = NP
3.Data structures

Answer: C · A stack

Why this is the answer

DFS explores as deep as possible, naturally using a stack (or recursion's call stack); BFS uses a queue.

Answer

C · A stack
4.Data structures

Answer: C · All records are in linked leaves, which makes range scans efficient

Why this is the answer

In a B+ tree internal nodes only guide the search and the leaves are linked in key order, so a range query finds the first key and then walks along the leaves.

Answer

C · All records are in linked leaves, which makes range scans efficient
5.Graph theory

Answer: k = 10

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.

Edges of a spanning tree

6 − 1 = 5

Edges to remove

k = 15 − 5 = 10

Answer

k = 10
6.Graph theory

Answer: W = 27

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

AC (1): take
BC (2): take
GC (3): take
AB (5): skip, closes a cycle
GE (6): take
DF (7): take
BD (8): take

Total weight

W = 1 + 2 + 3 + 6 + 7 + 8 = 27

Answer

W = 27

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