CF 102697036 - Speed Limits

The problem maintains a database of states. For every state, the database stores its maximum allowed speed. After building this database, we receive several queries, where each query names one state. For every queried state, we must print the speed limit stored for that state.

codeforcescompetitive-programming

CF 102697036 - Speed Limits

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Solution

Problem Understanding

The problem maintains a database of states. For every state, the database stores its maximum allowed speed. After building this database, we receive several queries, where each query names one state. For every queried state, we must print the speed limit stored for that state.

The key observation is that the queries are independent. Nothing changes in the database after a query, so the only task is to efficiently associate each state name with its stored integer.

The statement gives only that the number of states and the number of queries are positive integers, without explicit numerical upper bounds. The time limit is 1 second and the memory limit is 256 MB. That strongly favors a linear-time solution in the total input size. A method that scans the entire database for every query can require quadratic work when both parts of the input are large. A hash table gives expected constant-time lookup, making the total work proportional to the number of database entries and queries.

State names contain no spaces, which means each database entry can be read as two whitespace-separated tokens. The queried state name can likewise be read directly as one token.

There are a few cases where a careless implementation can fail. A repeated query must be answered repeatedly without modifying the database. For example,

2
Texas 85
Hawaii 60
3
Texas
Texas
Hawaii

produces

85
85
60

An implementation that accidentally removes a state after answering the first query would incorrectly fail on the second Texas.

A state with a one-character name is also valid because state names are simply strings without spaces. For example,

1
A 50
1
A

must produce

50

Code that assumes state names have a particular length would be unnecessarily fragile.

The database can also contain a state whose speed limit is zero if the input format permits it, and the value should be stored and printed exactly like any other integer. The correct approach does not attach special meaning to the numerical value.

Approaches

The direct brute-force approach is to store the database entries in an array. For every query, scan that array from the beginning until the requested state is found, then print its speed limit. This is correct because every database entry explicitly associates a state name with its speed limit, so finding the matching name gives exactly the required answer.

The problem is the repeated scanning. If there are (S) database entries and (T) queries, a query can inspect all (S) entries before finding its state. In the worst case this performs (S \times T) comparisons. If both values are around (100000), that is about (10^{10}) comparisons, far beyond what a 1 second time limit can support.

The useful observation is that the query key is the state name itself. We do not need the database to remain ordered, and we never need to search by a range or by a numeric property. We only need to answer exact equality queries of the form "what value belongs to this string?" A hash table is designed precisely for this operation.

We can store each pair as speed[state] = limit. Once the dictionary has been built, looking up a state takes expected (O(1)) time. Reading the database takes (O(S)), and answering all queries takes expected (O(T)). The brute-force method and the optimal method perform the same logical operation, but the dictionary changes how the matching state is found.

Approach Time Complexity Space Complexity Verdict
Brute Force (O(S T)) (O(S)) Too slow for large input
Hash Table Expected (O(S + T)) (O(S)) Accepted

Algorithm Walkthrough

  1. Read the number (S) of states in the database. We need this value only to know how many database records follow.
  2. Create an empty dictionary. The state name will be the key and its speed limit will be the associated value, because every future query gives us the state name.
  3. Read each of the (S) database records. Split the line into the state name and its integer speed limit, then store the pair in the dictionary. After this phase, every state can be found directly from its name.
  4. Read the number (T) of queries. Each following line contains exactly one state name.
  5. For every query, look up the state name in the dictionary and append the corresponding speed limit to the output. Appending answers and printing them together avoids unnecessary repeated output operations.
  6. Print all collected answers, one per line. The order is exactly the query order because answers are appended as the queries are processed.

Why it works

The invariant is that after processing the database, the dictionary contains the stored speed limit for every state in the input. When a query names a state, the dictionary lookup retrieves the value associated with exactly that name. Since the database is never modified while queries are being answered, repeated queries retrieve the same value each time. Thus every printed number is the speed limit belonging to the queried state.

Python Solution

import sys
input = sys.stdin.readline

def solve():
    s = int(input())
    speed = {}

    for _ in range(s):
        name, limit = input().split()
        speed[name] = int(limit)

    t = int(input())
    answer = []

    for _ in range(t):
        name = input().strip()
        answer.append(str(speed[name]))

    sys.stdout.write("\n".join(answer))

if __name__ == "__main__":
    solve()

The dictionary speed implements the mapping from state names to speed limits described in step 2. The database loop reads exactly (S) records and converts the numerical field to an integer before storing it.

The query loop reads exactly (T) names. strip() removes the newline at the end of each input line, leaving the state name used as the dictionary key. Since the state names contain no spaces, either strip() or split()[0] would work here.

The answers are stored as strings so that they can be joined with newline characters and written once. This is preferable to calling print for every query when the number of queries is large.

Python integers do not overflow, so there is no special handling needed for the speed limit value. The main implementation detail is preserving the exact state name as the dictionary key. Case is not normalized because state names in the input are identifiers, not case-insensitive text.

Worked Examples

For the first sample, the dictionary is built from the six database entries.

Query State Lookup result Output
1 NewYork 65 65
2 Pennsylvania 70 70
3 NewYork 65 65
4 Texas 85 85
5 Hawaii 60 60

The resulting output is 65, 70, 65, 85, and 60. The repeated NewYork query demonstrates why the database must remain unchanged between queries.

For a second example, consider a database containing only one state.

1
A 50
4
A
A
A
A

The dictionary contains one entry, A -> 50.

Query State Lookup result Output
1 A 50 50
2 A 50 50
3 A 50 50
4 A 50 50

The output is four lines containing 50. This trace confirms that the same database entry can answer an arbitrary number of queries without being consumed or altered.

Complexity Analysis

Measure Complexity Explanation
Time Expected (O(S + T)) Each database entry is inserted once and each query performs one expected constant-time hash lookup.
Space (O(S)) The dictionary stores one mapping for each database state.

The solution processes each input record a constant number of times. Even with very large values of (S) and (T), the amount of work grows linearly with the input rather than multiplying the two sizes. That is the appropriate complexity for the 1 second time limit and 256 MB memory limit.

Test Cases

import sys
import io

def solve():
    input = sys.stdin.readline

    s = int(input())
    speed = {}

    for _ in range(s):
        name, limit = input().split()
        speed[name] = int(limit)

    t = int(input())
    answer = []

    for _ in range(t):
        name = input().strip()
        answer.append(str(speed[name]))

    sys.stdout.write("\n".join(answer))

def run(inp: str) -> str:
    old_stdin = sys.stdin
    old_stdout = sys.stdout

    sys.stdin = io.StringIO(inp)
    sys.stdout = io.StringIO()

    try:
        solve()
        return sys.stdout.getvalue()
    finally:
        sys.stdin = old_stdin
        sys.stdout = old_stdout

# Provided sample
assert run(
    """6
NewYork 65
Texas 85
Pennsylvania 70
Massachussets 65
Washington 75
Hawaii 60
5
NewYork
Pennsylvania
NewYork
Texas
Hawaii
"""
) == "65\n70\n65\n85\n60", "sample 1"

# Minimum-size input
assert run(
    """1
A 1
1
A
"""
) == "1", "minimum-size case"

# All queries ask for the same state
assert run(
    """3
Alpha 10
Beta 20
Gamma 30
5
Beta
Beta
Beta
Beta
Beta
"""
) == "20\n20\n20\n20\n20", "repeated queries"

# Large input, generated maximum-style stress case
large_db = 100000
large_queries = 100000
large_input = [str(large_db)]

for i in range(large_db):
    large_input.append(f"S{i} {i}")

large_input.append(str(large_queries))

for i in range(large_queries):
    large_input.append(f"S{i}")

large_input = "\n".join(large_input) + "\n"

expected = "\n".join(str(i) for i in range(large_queries))
assert run(large_input) == expected, "large linear-time case"

# Boundary values and one-character names
assert run(
    """4
X 0
LongState 1000000000
Z 7
Middle 42
4
X
LongState
Z
Middle
"""
) == "0\n1000000000\n7\n42", "boundary values"
Test input Expected output What it validates
1 / A 1 / A 1 Minimum database and query size
Three states with five Beta queries Five lines containing 20 Repeated lookups do not modify the dictionary
100000 generated states and 100000 queries 0 through 99999 Linear scaling with a large input
Four states including limits 0 and 1000000000 0, 1000000000, 7, 42 Numerical boundaries and arbitrary state-name lengths

Edge Cases

The repeated-query case is handled naturally because dictionary lookup is read-only. For

2
Texas 85
Hawaii 60
3
Texas
Texas
Hawaii

the first Texas lookup returns 85, the second performs the same lookup and again returns 85, and Hawaii returns 60. The final output is 85, 85, 60. Nothing is removed from the dictionary after a lookup.

The minimum-size case has exactly one database record and one query:

1
A 1
1
A

The algorithm inserts A -> 1, then immediately retrieves 1. The output is 1. There is no special case in the implementation because a one-entry dictionary behaves exactly like a larger dictionary.

A zero speed limit does not require separate handling. For example,

2
X 0
Y 50
2
X
Y

creates the mappings X -> 0 and Y -> 50, so the output is

0
50

Using if speed[name]: instead of directly retrieving speed[name] could accidentally treat zero as missing, which is why the solution simply converts the stored value to a string and prints it.

Large numeric values are also handled directly. With

1
LongState 1000000000
1
LongState

the dictionary stores the integer without loss, and the output is 1000000000. Python's integer representation avoids overflow concerns.

Finally, a large number of queries does not increase memory proportionally to the number of queries in the solution. The dictionary contains the database, while the answer list contains the output strings. The lookup itself remains constant-time on average, so repeated or highly concentrated queries do not change the asymptotic running time.