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#729460#9520. Concave HullklhwongTL 634ms9528kbC++174.9kb2024-11-09 17:10:062024-11-09 17:10:07

Judging History

你现在查看的是最新测评结果

  • [2024-11-09 17:10:07]
  • 评测
  • 测评结果:TL
  • 用时:634ms
  • 内存:9528kb
  • [2024-11-09 17:10:06]
  • 提交

answer

#include <bits/stdc++.h>
#define eb emplace_back
using namespace std;
typedef long long ll;

// Structure to represent a point
struct P {
    ll x, y;
    P(ll x_=0, ll y_=0) : x(x_), y(y_) {}
    
    // Operator for sorting
    bool operator<(const P& other) const {
        return x < other.x || (x == other.x && y < other.y);
    }
};

// Cross product of OA and OB vectors
ll cross(const P& O, const P& A, const P& B) {
    ll dx1 = A.x - O.x;
    ll dy1 = A.y - O.y;
    ll dx2 = B.x - O.x;
    ll dy2 = B.y - O.y;
    return dx1 * dy2 - dx2 * dy1;
}

// Function to compute the convex hull using Andrew's algorithm
vector<P> convex_hull(vector<P> pts) {
    int n = pts.size();
    if(n == 0) return {};
    sort(pts.begin(), pts.end());
    // Remove duplicates if any (though problem states all points are distinct)
    pts.erase(unique(pts.begin(), pts.end(), [&](const P& a, const P& b) -> bool {
        return a.x == b.x && a.y == b.y;
    }), pts.end());
    
    n = pts.size();
    if(n == 1) return pts;
    
    vector<P> hull;
    // Lower hull
    for(int i = 0; i < n; ++i){
        while(hull.size() >=2 && cross(hull[hull.size()-2], hull[hull.size()-1], pts[i]) <= 0){
            hull.pop_back();
        }
        hull.eb(pts[i]);
    }
    // Upper hull
    int lower_size = hull.size();
    for(int i = n-2; i >=0 ; --i){
        while(hull.size() > lower_size && cross(hull[hull.size()-2], hull[hull.size()-1], pts[i]) <= 0){
            hull.pop_back();
        }
        hull.eb(pts[i]);
    }
    // Remove the last point because it's the same as the first one
    if(hull.size() >1) hull.pop_back();
    return hull;
}

// Function to compute twice the area of a polygon
ll polygon_area_twice(const vector<P>& poly){
    ll area = 0;
    int n = poly.size();
    for(int i=0;i<n;i++){
        int j = (i+1)%n;
        area += poly[i].x * poly[j].y - poly[j].x * poly[i].y;
    }
    return abs(area);
}

inline void work(){
    int n;
    cin >> n;
    vector<P> points(n);
    for(int i=0;i<n;i++) cin >> points[i].x >> points[i].y;
    
    // Compute Convex Hull CH1
    vector<P> CH1 = convex_hull(points);
    if((int)CH1.size() == n){
        // All points are on convex hull, cannot form concave polygon
        cout << "-1\n";
        return;
    }
    
    // Collect interior points
    // To efficiently find interior points, mark CH1 points
    // First, sort CH1 for binary search
    vector<P> sorted_CH1 = CH1;
    sort(sorted_CH1.begin(), sorted_CH1.end(), [&](const P& a, const P& b) -> bool {
        return a.x < b.x || (a.x == b.x && a.y < b.y);
    });
    // Collect interior points
    vector<P> interior;
    for(auto &pt : points){
        // Binary search in sorted_CH1
        if(!binary_search(sorted_CH1.begin(), sorted_CH1.end(), pt, [&](const P& a, const P& b) -> bool {
            return (a.x < b.x) || (a.x == b.x && a.y < b.y);
        })){
            interior.eb(pt);
        }
    }
    
    if(interior.empty()){
        // No interior points, cannot form concave polygon
        cout << "-1\n";
        return;
    }
    
    // Compute Convex Hull of interior points (CH2)
    vector<P> CH2 = convex_hull(interior);
    
    // To handle cases where CH2 has less than 3 points
    // Even with one or two points, we can still select them as potential x
    // So no special handling is needed since |ab cross (x -a)| can be computed for any x
    
    // Compute twice the area of CH1
    ll CH1_area_twice = polygon_area_twice(CH1);
    
    // Initialize minimum decrease in area
    ll min_decrease = LLONG_MAX;
    
    // Iterate over each edge ab in CH1
    int m = CH1.size();
    for(int i=0;i<m;i++){
        P a = CH1[i];
        P b = CH1[(i+1)%m];
        // For each edge ab, find the minimal |ab cross (x -a)| over all x in CH2
        // Since CH2 is convex, we can use ternary search to find the minimal value
        // However, the minimal value could be at any point, so it's safer to iterate all points in CH2
        // Given the constraints, this should be acceptable
        ll ab_dx = b.x - a.x;
        ll ab_dy = b.y - a.y;
        for(auto &x : CH2){
            ll cross_val = abs(ab_dx * (x.y - a.y) - ab_dy * (x.x - a.x));
            if(cross_val < min_decrease){
                min_decrease = cross_val;
                // Early exit if we find the minimal possible value
                if(min_decrease == 1) break;
            }
        }
        if(min_decrease == 1) break;
    }
    
    // Now, the maximum area of concave polygon is CH1_area_twice - min_decrease
    ll result = CH1_area_twice - min_decrease;
    // Since the problem asks to output twice the area, and we have already maintained it as twice the area
    cout << result << "\n";
}

int main(){
    cin.tie(0);
    ios::sync_with_stdio(false);
    int t;
    cin >> t;
    while(t--) work();
}

詳細信息

Test #1:

score: 100
Accepted
time: 0ms
memory: 3612kb

input:

2
6
-2 0
1 -2
5 2
0 4
1 2
3 1
4
0 0
1 0
0 1
1 1

output:

40
-1

result:

ok 2 lines

Test #2:

score: 0
Accepted
time: 1ms
memory: 3528kb

input:

10
243
-494423502 -591557038
-493438474 -648991734
-493289308 -656152126
-491185085 -661710614
-489063449 -666925265
-464265894 -709944049
-447472922 -737242534
-415977509 -773788538
-394263365 -797285016
-382728841 -807396819
-373481975 -814685302
-368242265 -818267002
-344482838 -833805545
-279398...

output:

2178418010787347715
1826413114144932145
1651687576234220014
1883871859778998985
2119126281997959892
894016174881844630
2271191316922158910
1998643358049669416
1740474221286618711
1168195646932543192

result:

ok 10 lines

Test #3:

score: 0
Accepted
time: 28ms
memory: 3752kb

input:

1000
125
64661186 -13143076
302828013 -185438065
-418713797 -191594241
430218126 -397441626
354327250 -836704374
149668812 -598584998
311305970 66790541
199720625 -592356787
468137 -584752683
258775829 96211747
-358669612 -134890109
-129221188 -998432368
-277309896 -140056561
356901185 420557649
-51...

output:

1986320445246155278
1900093336073022078
1612088392301142476
2012259136539173407
1819942017252118749
1772230185841892196
1164835025329039520
1527446155241140517
1807368432185303666
1236918659444944569
1306839249967484778
1984123720246784099
1868728080720036006
667458140583450322
2127932992585026491
4...

result:

ok 1000 lines

Test #4:

score: 0
Accepted
time: 39ms
memory: 3564kb

input:

10000
9
484630042 51929469
-40468396 -517784096
98214104 -103353239
629244333 -475172587
106398764 153884485
49211709 -44865749
1 10
166321833 -247717657
406208245 668933360
13
548702216 -631976459
37150086 -292461024
707804811 -486185860
239775286 -903166050
10096571 -541890068
686103484 558731937
...

output:

950319193795831919
1661025342421008544
1285164852091455548
1159924751776806668
1206071151805176722
794021230296144371
699991678992587791
1133990718508584290
1486311831172661605
984875884297072200
1327767982175057345
758247019006396699
1355381234262206155
1139262078529131471
1613462877860621700
12392...

result:

ok 10000 lines

Test #5:

score: 0
Accepted
time: 224ms
memory: 3984kb

input:

100
439
471536154 -312612104
155692036 -937312180
-461624056 -357636609
236656684 -911414873
-288656914 -74788431
-465779694 -381475149
-334197401 -903065737
491513067 -447615916
337664889 -852236281
-281689379 -53519178
-159101704 -920779200
-326159514 -95396204
21868593 -994282736
488425383 -41046...

output:

1973162724053130487
2054612790507830954
1726805687754843724
1699420177872986528
2129388571309147631
2198295137903288810
1697185883164440272
1219697450095721478
2027023581922285255
1674691247127206655
1673105966817209954
2179188692918747442
2146544318743443141
2230356305133660648
1676850321902993764
...

result:

ok 100 lines

Test #6:

score: 0
Accepted
time: 143ms
memory: 3840kb

input:

100
1362
-467257672 -466669
-467054869 -478108
-466973270 -481776
-466556983 -499770
-466329414 -508693
-466248017 -511805
-466158865 -513786
-466101273 -515035
-465927700 -518748
-465717624 -522106
-465303448 -528127
-465124548 -530726
-464649746 -536693
-464554872 -537799
-464478196 -538680
-46416...

output:

1666097696993497
1791366071767866
1806187278469532
1683419854733713
1685891971828916
1730190225081651
1787048201197565
1850308098208660
1710694884375502
1826363113637639
1816375352374659
2047431269497691
1549806516003854
1829438662895747
1678707854135065
1687423392883819
2121960009997761
16687219538...

result:

ok 100 lines

Test #7:

score: 0
Accepted
time: 634ms
memory: 6792kb

input:

2
62666
-486101704 -505730259
-486101698 -506082699
-486101689 -506111362
-486101682 -506126031
-486101528 -506293759
-486101259 -506556385
-486101196 -506613483
-486101154 -506648604
-486100935 -506831392
-486100631 -507083675
-486100470 -507199151
-486100233 -507368923
-486100193 -507397039
-48609...

output:

2178736946152219010
1825181940245096152

result:

ok 2 lines

Test #8:

score: 0
Accepted
time: 532ms
memory: 9528kb

input:

2
100000
301945097 76373292
467957663 -286424714
8245445 -597212507
-474204621 -708828667
184159460 105942538
443435905 -429212625
490658771 -382198656
82512047 -612522436
-228221388 -965890088
394789011 -145801151
-106120174 -528202395
428939626 -194437311
497429477 -527407728
365739746 -114818962
...

output:

2502889432701099511
2267250485735988121

result:

ok 2 lines

Test #9:

score: -100
Time Limit Exceeded

input:

2
100000
221128057 -975244780
-618765360 -785575858
422567455 -906331476
-988680318 -150037424
-929870145 367887908
-757813541 -652471177
291995621 -956419655
-785381507 619012026
468864522 -883270094
-588416522 808557973
859345881 511394814
988105866 153775152
216931298 -976186873
467050734 8842305...

output:


result: