void solve (int x) {
ans.clear();
ans.l1 = 0;
ans.l2 = t;
ans.a[0][0] = 1;
f.l1 = f.l2 = t;
//f.print();
while (x) {
if (x % 2) ans = ans * f;
f = f * f;
x = x/2;
}
//ans = ans * f;
}
void reparametrize_func(const vec & lmk, vec & lnew, mat & LNEW_lmk) const {
vec7 lmk1;
vec7 lmk2;
vec3 euc1;
vec3 euc2;
mat EUC1_lmk1(3,7);
mat EUC2_lmk2(3,7);
LNEW_lmk.clear();
subrange(lmk1,0,3) = subrange(lmk,0,3);
subrange(lmk1,3,7) = subrange(lmk,3,7);
subrange(lmk2,0,3) = subrange(lmk,0,3);
subrange(lmk2,3,7) = subrange(lmk,7,11);
lmkAHP::ahp2euc(lmk1,euc1,EUC1_lmk1);
lmkAHP::ahp2euc(lmk2,euc2,EUC2_lmk2);
subrange(lnew,0,3) = euc1;
subrange(lnew,3,6) = euc2;
subrange(LNEW_lmk,0,3,0,7) = EUC1_lmk1;
subrange(LNEW_lmk,3,6,0,3) = subrange(EUC2_lmk2,0,3,0,3);
subrange(LNEW_lmk,3,6,7,11) = subrange(EUC2_lmk2,0,3,3,7);
}
int main () {
freopen ("sequence.in", "r", stdin);
freopen ("sequence.out", "w", stdout);
f.clear();
cin >> m >> n;
for (int i=1; i<=m; i++) scanf("%s", str[i]);
for (int i=1; i<=m; i++)
for (int j=0; str[i][j]; j++)
s[i] += str[i][j];
for (int i=1; i<=m; i++)
for (int j=1; j<=m; j++)
if (i != j && sub (i, j))
del[i] = 1;
for (int i=1; i<=m; i++)
if (!del[i]) {
int now = 0;
for (int j=0; j<s[i].size(); j++) {
if (trie[now].mark) break;
if (!trie[now].a[turn(s[i][j])])
trie[now].a[turn(s[i][j])] = ++t;
now = trie[now].a[turn(s[i][j])];
}
trie[now].mark = 1;
}
int q[101], head = 0, tail = 0, now = 0;
memset (q , 0, sizeof (q));
for (int i=0; i<4; i++)
if (now = trie[0].a[i]) {
if (!trie[now].mark) {
f.a[0][now] ++;
q[tail] = now;
tail++;
}
}
else f.a[0][0] ++;
while (head < tail) {
for (int i=0; i<4; i++)
if (trie[q[head]].a[i] && !trie[trie[q[head]].a[i]].mark) {
int now = trie[q[head]].fail;
while (now && !trie[now].a[i])
now = trie[now].fail;
trie[trie[q[head]].a[i]].fail = trie[now].a[i];
q[tail] = trie[q[head]].a[i];
tail++;
}
head++;
}
for (int i=1; i<=t; i++)
for (int j=0; j<4; j++)
if (!trie[i].mark) {
int cur = i;
while (cur && !trie[cur].a[j]) cur = trie[cur].fail;
if (!trie[trie[cur].a[j]].mark)
f.a[i][trie[cur].a[j]]++;
}
//ans = f;
//for (int i=1; i<n; i++)
// ans = mul (ans, f);
solve (n);
for (int i=0; i<=t; i++)
an += ans.a[0][i];
cout << an % 100000 << endl;
}
/*
* This motion model is driven by IMU measurements and random perturbations, and defined by:
* The state vector, x = [p q v ab wb g] , of size 19.
*
* The transition equation is
* - x+ = move_func(x,u,n),
*
* with u = [am, wm] the IMU measurements (the control input)
* and n = [vi, ti, abi, wbi] the perturbation impulse.
*
* the transition equation f() is decomposed as:
* #if AVGSPEED
* - p+ = p + (v + v+)/2*dt
* #else
* - p+ = p + v*dt
* #endif
* - q+ = q**((wm - wb)*dt + ti) <-- ** : quaternion product ; ti : theta impulse
* - v+ = v + (R(q)*(am - ab) + g)*dt + vi <-- am and wm: IMU measurements ; vi : v impulse
* - ab+ = ab + abi <-- abi : random walk in acc bias with abi impulse perturbation
* - wb+ = wb + wbi <-- wbi : random walk of gyro bias with wbi impulse perturbation
* - g+ = g <-- g : gravity vector, constant but unknown
* -----------------------------------------------------------------------------
*
* The Jacobian XNEW_x is built with
* var | p q v ab wb g
* pos | 0 3 7 10 13 16
* -------+----------------------------------------------------
* #if AVGSPEED
* p 0 | I VNEW_q*dt/2 I*dt -R*dt*dt/2 0 I*dt*dt/2
* #else
* p 0 | I 0 I*dt 0 0 0
* #endif
* q 3 | 0 QNEW_q 0 0 QNEW_wb 0
* v 7 | 0 VNEW_q I -R*dt 0 I*dt
* ab 10 | 0 0 0 I 0 0
* wb 13 | 0 0 0 0 I 0
* g 16 | 0 0 0 0 0 I
* -----------------------------------------------------------------------------
*
* The Jacobian XNEW_pert is built with
* var | vi ti abi wbi
* pos | 0 3 6 9
* -------+----------------------
* #if AVGSPEED
* p 0 |I.dt/2 0 0 0
* #else
* p 0 | 0 0 0 0
* #endif
* q 3 | 0 QNEW_ti 0 0
* v 7 | I 0 0 0
* ab 10 | 0 0 I 0
* wb 13 | 0 0 0 I
* g 16 | 0 0 0 0
* -----------------------------------------------------------------------------
*/
void RobotInertial::move_func(const vec & _x, const vec & _u, const vec & _n, double _dt, vec & _xnew, mat & _XNEW_x,
mat & _XNEW_pert) {
// Separate things out to make it clearer
vec3 p, v, ab, wb, gv;
vec g;
vec4 q;
splitState(_x, p, q, v, ab, wb, g); // split state vector
if (g_size == 1) { gv = z_axis * g(0); } else { gv = g; }
// Split control and perturbation vectors into
// sensed acceleration and sensed angular rate
// and noises
vec3 am, wm, vi, ti, abi, wbi; // measurements and random walks
splitControl(_u, am, wm);
splitPert(_n, vi, ti, abi, wbi);
// It is useful to start obtaining a nice rotation matrix and the product R*dt
Rold = q2R(q);
Rdt = Rold * _dt;
// Invert sensor functions. Get true acc. and ang. rates
// a = R(q)(asens - ab) + g true acceleration
// w = wsens - wb true angular rate
vec3 atrue, wtrue;
atrue = prod(Rold, (am - ab)) + gv; // could have done rotate(q, instead of prod(Rold,
wtrue = wm - wb;
// Get new state vector
vec3 pnew, vnew, abnew, wbnew;
vec gnew;
vec4 qnew;
// qnew = q x q(w * dt)
// Keep qwt ( = q(w * dt)) for later use
vec4 qwdt = v2q(wtrue * _dt + ti);
qnew = qProd(q, qwdt); // orientation
vnew = v + atrue * _dt + vi; // velocity
#if AVGSPEED
pnew = p + (v+vnew)/2 * _dt; // position
#else
pnew = p + v * _dt; // position
#endif
abnew = ab + abi; // acc bias
wbnew = wb + wbi; // gyro bias
gnew = g; // gravity does not change
// normalize quaternion
ublasExtra::normalizeJac(qnew, QNORM_qnew);
ublasExtra::normalize(qnew);
// Put it all together - this is the output state
unsplitState(pnew, qnew, vnew, abnew, wbnew, gnew, _xnew);
// Now on to the Jacobian...
// Identity is a good place to start since overall structure is like this
// var | p q v ab wb g
// pos | 0 3 7 10 13 16
// -------+----------------------------------------------------
//#if AVGSPEED
// p 0 | I VNEW_q*dt/2 I*dt -R*dt*dt/2 0 I*dt*dt/2
//#else
// p 0 | I 0 I*dt 0 0 0
//#endif
// q 3 | 0 QNEW_q 0 0 QNEW_wb 0
// v 7 | 0 VNEW_q I -R*dt 0 I*dt
// ab 10 | 0 0 0 I 0 0
// wb 13 | 0 0 0 0 I 0
// g 16 | 0 0 0 0 0 I
_XNEW_x.assign(identity_mat(state.size()));
// Fill in XNEW_v: VNEW_g and PNEW_v = I * dt
identity_mat I(3);
Idt = I * _dt;
mat Iz; if (g_size == 1) { Iz.resize(3,1); Iz.clear(); Iz(2,0)=1; } else { Iz = I; }
mat Izdt = Iz * _dt;
subrange(_XNEW_x, 0, 3, 7, 10) = Idt;
#if AVGSPEED
subrange(_XNEW_x, 0, 3, 16, 16+g_size) = Izdt*_dt/2;
#endif
subrange(_XNEW_x, 7, 10, 16, 16+g_size) = Izdt;
// Fill in QNEW_q
// qnew = qold ** qwdt ( qnew = q1 ** q2 = qProd(q1, q2) in rtslam/quatTools.hpp )
qProd_by_dq1(qwdt, QNEW_q);
subrange(_XNEW_x, 3, 7, 3, 7) = prod(QNORM_qnew, QNEW_q);
// Fill in QNEW_wb
// QNEW_wb = QNEW_qwdt * QWDT_wdt * WDT_w * W_wb
// = QNEW_qwdt * QWDT_w * W_wb
// = QNEW_qwdt * QWDT_w * (-1)
qProd_by_dq2(q, QNEW_qwdt);
// Here we get the derivative of qwdt wrt wtrue, so we consider dt = 1 and call for the derivative of v2q() with v = w*dt
// v2q_by_dv(wtrue, QWDT_w);
v2q_by_dv(wtrue*_dt, QWDT_w); QWDT_w *= _dt;
QNEW_w = prod ( QNEW_qwdt, QWDT_w);
subrange(_XNEW_x, 3, 7, 13, 16) = -prod(QNORM_qnew,QNEW_w);
// Fill VNEW_q
// VNEW_q = d(R(q)*v) / dq
rotate_by_dq(q, am-ab, VNEW_q); VNEW_q *= _dt;
subrange(_XNEW_x, 7, 10, 3, 7) = VNEW_q;
#if AVGSPEED
subrange(_XNEW_x, 0, 3, 3, 7) = VNEW_q*_dt/2;
#endif
// Fill in VNEW_ab
subrange(_XNEW_x, 7, 10, 10, 13) = -Rdt;
#if AVGSPEED
subrange(_XNEW_x, 0, 3, 10, 13) = -Rdt*_dt/2;
#endif
// Now on to the perturbation Jacobian XNEW_pert
// Form of Jacobian XNEW_pert
// It is like this:
// var | vi ti abi wbi
// pos | 0 3 6 9
// -------+----------------------
//#if AVGSPEED
// p 0 |I.dt/2 0 0 0
//#else
// p 0 | 0 0 0 0
//#endif
// q 3 | 0 QNEW_ti 0 0
// v 7 | I 0 0 0
// ab 10 | 0 0 I 0
// wb 13 | 0 0 0 I
// g 16 | 0 0 0 0
// Fill in the easy bits first
_XNEW_pert.clear();
#if AVGSPEED
ublas::subrange(_XNEW_pert, 0, 3, 0, 3) = Idt/2;
#endif
ublas::subrange(_XNEW_pert, 7, 10, 0, 3) = I;
ublas::subrange(_XNEW_pert, 10, 13, 6, 9) = I;
ublas::subrange(_XNEW_pert, 13, 16, 9, 12) = I;
// Tricky bit is QNEW_ti = d(qnew)/d(ti)
// Here, ti is the integral of the perturbation, ti = integral_{tau=0}^dt (wn(t) * dtau),
// with: wn: the angular rate measurement noise
// dt: the integration period
// ti: the resulting angular impulse
// The integral of the dynamic equation is:
// q+ = q ** v2q((wm - wb)*dt + ti)
// We have: QNEW_ti = QNEW_w / dt
// with: QNEW_w computed before.
// The time dependence needs to be included in perturbation.P(), proportional to perturbation.dt:
// U = perturbation.P() = U_continuous_time * dt
// with: U_continuous_time expressed in ( rad / s / sqrt(s) )^2 = rad^2 / s^3 <-- yeah, it is confusing, but true.
// (Use perturbation.set_P_from_continuous() helper if necessary.)
//
subrange(_XNEW_pert, 3, 7, 3, 6) = prod (QNORM_qnew, QNEW_w) * (1 / _dt);
}
void RobotCenteredConstantVelocity::move_func(
const vec & _x, const vec & _u,
const vec & _n, const double _dt,
vec & _xnew, mat & _XNEW_x,
mat & _XNEW_u) {
using namespace jblas;
using namespace ublas;
/*
*
* The state vector, x = [p q v w pBase qBase] = [x y z, qw qx qy qz, vx vy vz, wx wy wz, xb yb zb, qwb qxb qyb qzb], of size 20.
* The transition equation x+ = move(x,i), with i = [vi wi] the control impulse, decomposed as:
*
* This motion model is defined by 2 steps :
* -----------------------------------------
* 1) We perform a frame transform of all the map :
* -a) The Landmarks
* From frame F0 to frame [p,q]
* -b) The Base frame
* From frame F0 to frame [p,q]
* -c) The Current robot pose become (we don't have to code this part because we will update the pose in step 2)
* p=0
* q=[1,0,0,0]'
* v=v
* w=w
*
* 2) We update the current pose with a constant velocity model :
* * p = v*dt
* * q = w2q(w*dt) <-- w2q : 3 angles to quaternion
* * v = v + vi <-- vi : impulse in linear velocity - vi = [vix viy viz]
* * w = w + wi <-- wi : impulse in angular velocity - wi = [wix wiy wiz]
*
* -----------------------------------------------------------------------------
*
* We have some jacobians :
* The Jacobian XNEW_x is built with
*
* _______________________ROBOT STATE______________________________
* / \
* p q v w pBase qBase lold |
* 0 3 7 10 13 16 20 |
* -------------------------------------------------------------------------------------------+------
* ____________________________________________________________________
* XNEW_x = [ | PNEW_v | ] | 0 p \|
* [ | QNEW_w | ] | 3 q |
* [ | I_3 | ] | 7 v | Robot State
* [ | I_3 | ] | 10 w |
* [ | [-----PBASENEW_f----] PBASENEW_pbase | ] | 13 pBase |
* [ | QBASENEW_q QBASENEW_qbase | ] | 16 qBase /|
* [ |__________________________________________________________________| ] |
* [ LNEW_p LNEW_q LNEW_lold ] | 20 lnew
*
* -----------------------------------------------------------------------------
*
* The Jacobian XNEW_pert is built with
* vi wi |
* 0 3 |
* -----------------+------
* ____________
* XNEW_pert = [| |] | 0 p \|
* [| |] | 3 q |
* [| I_3 |] | 7 v | Robot State
* [| I_3 |] | 10 w |
* [| |] | 7 pBase |
* [|____________|] | 10 qBase /|
* [ ] | 7 lnew
* this Jacobian is however constant and is computed once at Construction time.
*
* NOTE: The also constant perturbation matrix:
* Q = XNEW_pert * perturbation.P * trans(XNEW_pert)
* could be built also once after construction with computeStatePerturbation().
* This is up to the user -- if nothing is done, Q will be computed at each iteration.
* -----------------------------------------------------------------------------
*/
lastJump = ublas::subrange(_x, 0, 7) ; // will be used in the reframe process of : the robot, the landmarks.
// TODO the internals jacobians (of fix size)
// can be defined in the class definition,
// or the RobotAbstract class.
// used variables :
// 1- robot at time t
vec3 p, v, w, pbase;
vec4 q, qbase;
// 2- robot at time t+1
vec3 pnew, vnew, wnew, pbasenew;
vec4 qnew, qbasenew;
// 3- jacobians from t to t+1
mat PBASENEW_f(3, 7);
mat33 PBASENEW_pbase ;
mat44 QBASENEW_q, QBASENEW_qbase;
identity_mat I_3(3);
// split robot state vector (F is the reference frame change between t and t+1)
splitState(_x, p, q, v, w, pbase, qbase);
lastJump = ublas::subrange(_x, 0, 7) ;
// split perturbation vector
vec3 vi, wi;
splitControl(_n, vi, wi);
// predict each part of the state, give or build non-trivial Jacobians
// 1- PQVW at t+1
pnew = v * _dt; // FIXME reframe the new position and quaternion (position R_t relative to frame R_t-1)
PNEW_v = I_3 * _dt;
quaternion::v2q(w * _dt, qnew, QNEW_wdt); // FiXME _w or _wdt
vnew = v + vi;
wnew = w + wi;
// 2- PQ-of-Base at t+1 (reframe the base frame)
quaternion::eucToFrame(lastJump, pbase, pbasenew, PBASENEW_f, PBASENEW_pbase) ; // pbase
quaternion::qProd(qbase, q, qbasenew, QBASENEW_qbase, QBASENEW_q) ; // qbase
// Re-compose state - this is the output state.
unsplitState(pnew, qnew, vnew, wnew, pbasenew, qbasenew, _xnew); // Robot state
// Build transition Jacobian matrix XNEW_x
_XNEW_x.clear() ;
project(_XNEW_x, range(0 , 3 ), range(7 , 10)) = PNEW_v ;
project(_XNEW_x, range(3 , 7 ), range(10, 13)) = QNEW_wdt ; // FIXME jacobian wrt w or wdt
project(_XNEW_x, range(7 , 10), range(7 , 10)) = I_3 ;
project(_XNEW_x, range(10, 13), range(10, 13)) = I_3 ;
project(_XNEW_x, range(13, 16), range(0 , 7 )) = PBASENEW_f ;
project(_XNEW_x, range(13, 16), range(13, 16)) = PBASENEW_pbase;
project(_XNEW_x, range(16, 20), range(3 , 7 )) = QBASENEW_q ;
project(_XNEW_x, range(16, 20), range(16, 20)) = QBASENEW_qbase;
// NOW the landmarks are defined in the frame F = [{0,0,0, 1,0,0,0} - lastJump],
// so we will reframe all the landmarks.
/*
* We are normally supposed here to build the perturbation Jacobian matrix XNEW_pert.
* NOTE: XNEW_pert is constant and it has been build in the constructor.
*
* \sa See computePertJacobian() for more info.
*/
}